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Properties

Label 1-323-323.314-r1-0-0
Degree $1$
Conductor $323$
Sign $0.997 - 0.0753i$
Analytic cond. $34.7111$
Root an. cond. $34.7111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

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

Invariants

Degree: \(1\)
Conductor: \(323\)    =    \(17 \cdot 19\)
Sign: $0.997 - 0.0753i$
Analytic conductor: \(34.7111\)
Root analytic conductor: \(34.7111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{323} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 323,\ (1:\ ),\ 0.997 - 0.0753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.506828502 - 0.05683926496i\)
\(L(\frac12)\) \(\approx\) \(1.506828502 - 0.05683926496i\)
\(L(1)\) \(\approx\) \(0.9493377612 - 0.3334732555i\)
\(L(1)\) \(\approx\) \(0.9493377612 - 0.3334732555i\)

Euler product

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

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 $Z$-function along the critical line