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Properties

Label 2-3276-91.34-c1-0-37
Degree 22
Conductor 32763276
Sign 0.700+0.713i0.700 + 0.713i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.09 − 2.09i)5-s + (2.62 + 0.311i)7-s + (−1.65 + 1.65i)11-s + (1.34 − 3.34i)13-s − 1.34·17-s + (0.481 − 0.481i)19-s + 5.63i·23-s − 3.78i·25-s + 5.32·29-s + (−0.730 + 0.730i)31-s + (6.16 − 4.85i)35-s + (5.37 − 5.37i)37-s + (6.40 − 6.40i)41-s − 8.68i·43-s + (6.94 + 6.94i)47-s + ⋯
L(s)  = 1  + (0.937 − 0.937i)5-s + (0.993 + 0.117i)7-s + (−0.497 + 0.497i)11-s + (0.373 − 0.927i)13-s − 0.325·17-s + (0.110 − 0.110i)19-s + 1.17i·23-s − 0.757i·25-s + 0.988·29-s + (−0.131 + 0.131i)31-s + (1.04 − 0.820i)35-s + (0.882 − 0.882i)37-s + (1.00 − 1.00i)41-s − 1.32i·43-s + (1.01 + 1.01i)47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.700+0.713i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((0.700+0.713i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.700+0.713i0.700 + 0.713i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(1945,⋅)\chi_{3276} (1945, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), 0.700+0.713i)(2,\ 3276,\ (\ :1/2),\ 0.700 + 0.713i)

Particular Values

L(1)L(1) ≈\approx 2.5807421982.580742198
L(12)L(\frac12) ≈\approx 2.5807421982.580742198
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
7 1+(−2.62−0.311i)T 1 + (-2.62 - 0.311i)T
13 1+(−1.34+3.34i)T 1 + (-1.34 + 3.34i)T
good5 1+(−2.09+2.09i)T−5iT2 1 + (-2.09 + 2.09i)T - 5iT^{2}
11 1+(1.65−1.65i)T−11iT2 1 + (1.65 - 1.65i)T - 11iT^{2}
17 1+1.34T+17T2 1 + 1.34T + 17T^{2}
19 1+(−0.481+0.481i)T−19iT2 1 + (-0.481 + 0.481i)T - 19iT^{2}
23 1−5.63iT−23T2 1 - 5.63iT - 23T^{2}
29 1−5.32T+29T2 1 - 5.32T + 29T^{2}
31 1+(0.730−0.730i)T−31iT2 1 + (0.730 - 0.730i)T - 31iT^{2}
37 1+(−5.37+5.37i)T−37iT2 1 + (-5.37 + 5.37i)T - 37iT^{2}
41 1+(−6.40+6.40i)T−41iT2 1 + (-6.40 + 6.40i)T - 41iT^{2}
43 1+8.68iT−43T2 1 + 8.68iT - 43T^{2}
47 1+(−6.94−6.94i)T+47iT2 1 + (-6.94 - 6.94i)T + 47iT^{2}
53 1+2.01T+53T2 1 + 2.01T + 53T^{2}
59 1+(−1.55−1.55i)T+59iT2 1 + (-1.55 - 1.55i)T + 59iT^{2}
61 1+0.0367iT−61T2 1 + 0.0367iT - 61T^{2}
67 1+(7.90+7.90i)T+67iT2 1 + (7.90 + 7.90i)T + 67iT^{2}
71 1+(−0.368−0.368i)T+71iT2 1 + (-0.368 - 0.368i)T + 71iT^{2}
73 1+(−1.66−1.66i)T+73iT2 1 + (-1.66 - 1.66i)T + 73iT^{2}
79 1+2.49T+79T2 1 + 2.49T + 79T^{2}
83 1+(5.60−5.60i)T−83iT2 1 + (5.60 - 5.60i)T - 83iT^{2}
89 1+(2.09+2.09i)T+89iT2 1 + (2.09 + 2.09i)T + 89iT^{2}
97 1+(−5.68+5.68i)T−97iT2 1 + (-5.68 + 5.68i)T - 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.664296468543077472426778665178, −7.81477640334811300267357600552, −7.27655825075583342152895153668, −5.94440808036611751386174744073, −5.51306568359767582510326261685, −4.87036896248186085582298999831, −4.04271037074655031487825440133, −2.69019440747933498839555317433, −1.82751226291855364772717915278, −0.897935021950691511777538779129, 1.19003431218260744205180064134, 2.29329633500258624806870933068, 2.88091921224217103772758102500, 4.20552654872162426336159241265, 4.84852379478817893917514187681, 5.93535567484433283523358677773, 6.37973993882686576560001945356, 7.16092455906109093279886495727, 8.075849395016884628059834353174, 8.644677484388445594964271632884

Graph of the ZZ-function along the critical line