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Properties

Label 2-3276-13.12-c1-0-31
Degree 22
Conductor 32763276
Sign −0.929+0.368i-0.929 + 0.368i
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  − 2.32i·5-s − i·7-s − 0.908i·11-s + (−1.32 − 3.35i)13-s − 1.93·17-s + 4.19i·19-s + 1.41·23-s − 0.419·25-s + 7.75·29-s − 3.87i·31-s − 2.32·35-s − 11.9i·37-s − 6.19i·41-s − 6.91·43-s + 3.16i·47-s + ⋯
L(s)  = 1  − 1.04i·5-s − 0.377i·7-s − 0.273i·11-s + (−0.368 − 0.929i)13-s − 0.468·17-s + 0.961i·19-s + 0.295·23-s − 0.0838·25-s + 1.44·29-s − 0.695i·31-s − 0.393·35-s − 1.97i·37-s − 0.968i·41-s − 1.05·43-s + 0.462i·47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((−0.929+0.368i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((−0.929+0.368i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −0.929+0.368i-0.929 + 0.368i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(2521,⋅)\chi_{3276} (2521, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), −0.929+0.368i)(2,\ 3276,\ (\ :1/2),\ -0.929 + 0.368i)

Particular Values

L(1)L(1) ≈\approx 1.1205372001.120537200
L(12)L(\frac12) ≈\approx 1.1205372001.120537200
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+iT 1 + iT
13 1+(1.32+3.35i)T 1 + (1.32 + 3.35i)T
good5 1+2.32iT−5T2 1 + 2.32iT - 5T^{2}
11 1+0.908iT−11T2 1 + 0.908iT - 11T^{2}
17 1+1.93T+17T2 1 + 1.93T + 17T^{2}
19 1−4.19iT−19T2 1 - 4.19iT - 19T^{2}
23 1−1.41T+23T2 1 - 1.41T + 23T^{2}
29 1−7.75T+29T2 1 - 7.75T + 29T^{2}
31 1+3.87iT−31T2 1 + 3.87iT - 31T^{2}
37 1+11.9iT−37T2 1 + 11.9iT - 37T^{2}
41 1+6.19iT−41T2 1 + 6.19iT - 41T^{2}
43 1+6.91T+43T2 1 + 6.91T + 43T^{2}
47 1−3.16iT−47T2 1 - 3.16iT - 47T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1−4.97iT−59T2 1 - 4.97iT - 59T^{2}
61 1+0.0483T+61T2 1 + 0.0483T + 61T^{2}
67 1+3.76iT−67T2 1 + 3.76iT - 67T^{2}
71 1+15.3iT−71T2 1 + 15.3iT - 71T^{2}
73 1−7.16iT−73T2 1 - 7.16iT - 73T^{2}
79 1+17.3T+79T2 1 + 17.3T + 79T^{2}
83 1−12.1iT−83T2 1 - 12.1iT - 83T^{2}
89 1−11.7iT−89T2 1 - 11.7iT - 89T^{2}
97 1−6.37iT−97T2 1 - 6.37iT - 97T^{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

−8.264029854260073140592375652281, −7.77368964572353663476173763135, −6.84762157409560247997729733270, −5.92111355338624973191352996807, −5.23550705277991893029180770264, −4.49490076306865935026856107007, −3.67063405331076510340095833844, −2.60973049736296790119946064418, −1.37637974374407496676731427861, −0.34562550069372353586130543541, 1.54774990506521917859998408974, 2.69180527314960563627560834455, 3.15595059815724933436593019100, 4.53520419526360059397432771017, 4.93755364249070895421740923103, 6.29549377461321243007543112424, 6.70642456043420696992974056442, 7.24140458977580950617048636838, 8.348651266755775118996114309817, 8.868890778665855734657795536641

Graph of the ZZ-function along the critical line