Properties

Label 1-323-323.314-r1-0-0
Degree 11
Conductor 323323
Sign 0.9970.0753i0.997 - 0.0753i
Analytic cond. 34.711134.7111
Root an. cond. 34.711134.7111
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.819 − 0.573i)3-s + (−0.766 + 0.642i)4-s + (0.0871 + 0.996i)5-s + (−0.819 − 0.573i)6-s + (−0.965 − 0.258i)7-s + (0.866 + 0.5i)8-s + (0.342 − 0.939i)9-s + (0.906 − 0.422i)10-s + (−0.258 − 0.965i)11-s + (−0.258 + 0.965i)12-s + (0.173 + 0.984i)13-s + (0.0871 + 0.996i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.819 − 0.573i)3-s + (−0.766 + 0.642i)4-s + (0.0871 + 0.996i)5-s + (−0.819 − 0.573i)6-s + (−0.965 − 0.258i)7-s + (0.866 + 0.5i)8-s + (0.342 − 0.939i)9-s + (0.906 − 0.422i)10-s + (−0.258 − 0.965i)11-s + (−0.258 + 0.965i)12-s + (0.173 + 0.984i)13-s + (0.0871 + 0.996i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + ⋯

Functional equation

Λ(s)=(323s/2ΓR(s+1)L(s)=((0.9970.0753i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(323s/2ΓR(s+1)L(s)=((0.9970.0753i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 323323    =    171917 \cdot 19
Sign: 0.9970.0753i0.997 - 0.0753i
Analytic conductor: 34.711134.7111
Root analytic conductor: 34.711134.7111
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ323(314,)\chi_{323} (314, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 323, (1: ), 0.9970.0753i)(1,\ 323,\ (1:\ ),\ 0.997 - 0.0753i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.5068285020.05683926496i1.506828502 - 0.05683926496i
L(12)L(\frac12) \approx 1.5068285020.05683926496i1.506828502 - 0.05683926496i
L(1)L(1) \approx 0.94933776120.3334732555i0.9493377612 - 0.3334732555i
L(1)L(1) \approx 0.94933776120.3334732555i0.9493377612 - 0.3334732555i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad17 1 1
19 1 1
good2 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
3 1+(0.8190.573i)T 1 + (0.819 - 0.573i)T
5 1+(0.0871+0.996i)T 1 + (0.0871 + 0.996i)T
7 1+(0.9650.258i)T 1 + (-0.965 - 0.258i)T
11 1+(0.2580.965i)T 1 + (-0.258 - 0.965i)T
13 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
23 1+(0.0871+0.996i)T 1 + (-0.0871 + 0.996i)T
29 1+(0.422+0.906i)T 1 + (0.422 + 0.906i)T
31 1+(0.2580.965i)T 1 + (0.258 - 0.965i)T
37 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
41 1+(0.573+0.819i)T 1 + (0.573 + 0.819i)T
43 1+(0.642+0.766i)T 1 + (-0.642 + 0.766i)T
47 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
53 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
59 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
61 1+(0.08710.996i)T 1 + (0.0871 - 0.996i)T
67 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
71 1+(0.0871+0.996i)T 1 + (0.0871 + 0.996i)T
73 1+(0.8190.573i)T 1 + (0.819 - 0.573i)T
79 1+(0.5730.819i)T 1 + (-0.573 - 0.819i)T
83 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
89 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
97 1+(0.906+0.422i)T 1 + (0.906 + 0.422i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−25.270938483773857166574974679163, −24.37624216092381160662363621231, −23.11896199492205938969758527893, −22.460074728353832516072067237806, −21.271328350167696643547687343274, −20.19056512151142350175262809905, −19.68323459954792705122820783875, −18.58393724817521353199410402987, −17.49712507426034271081959676997, −16.5378243855636843785134194962, −15.73784686027720654810650058380, −15.292941488405112639543601570588, −14.11149061643311918869092332456, −13.1148082405760171512609522753, −12.51480464973733815250919543138, −10.35093655857758582569483644616, −9.81490682307374492273813693314, −8.87696858522090591191337867386, −8.20549035155178899433021003395, −7.12802264732261221374052311851, −5.7683278068157574512104597621, −4.841855382803566039660492070237, −3.85278306952365046957986912304, −2.27931556101184990341272905514, −0.52197493142318869995594297961, 1.070581788946561308561063357628, 2.45932431358397884268827853620, 3.19258908505975175220230531535, 4.00725518470116078726958549904, 6.1132720720620890634409631822, 7.10673568359213952955860137419, 8.08537201976496821554213253998, 9.22314279246485052984449068842, 9.8944095504244044709852088639, 11.02770615454567631269940752687, 11.897587344356576185170486672185, 13.16522762733238226597196336909, 13.6533196137855694132969448539, 14.48334491262920641473465425020, 15.84239360757113224165937012919, 16.97971371620191585084482836710, 18.237847917566086847948091353480, 18.781903290241693894296843492492, 19.396426676683406310410199450581, 20.12559919990118624105625007973, 21.384470681784750658787863527849, 21.86540167062890737203965828021, 23.07075891889905636527923536474, 23.784456341130253679067509789133, 25.25247096670516008109792720430

Graph of the ZZ-function along the critical line