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Properties

Label 2-3380-3380.3299-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.976 - 0.215i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
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  + (−0.845 − 0.534i)2-s + (0.428 + 0.903i)4-s + (−0.948 − 0.316i)5-s + (0.120 − 0.992i)8-s + (0.0402 − 0.999i)9-s + (0.632 + 0.774i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (−1.08 + 1.44i)17-s + (−0.568 + 0.822i)18-s + (−0.120 − 0.992i)20-s + (0.799 + 0.600i)25-s + (0.996 − 0.0804i)26-s + (0.470 + 0.297i)29-s + (0.948 − 0.316i)32-s + ⋯
L(s)  = 1  + (−0.845 − 0.534i)2-s + (0.428 + 0.903i)4-s + (−0.948 − 0.316i)5-s + (0.120 − 0.992i)8-s + (0.0402 − 0.999i)9-s + (0.632 + 0.774i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (−1.08 + 1.44i)17-s + (−0.568 + 0.822i)18-s + (−0.120 − 0.992i)20-s + (0.799 + 0.600i)25-s + (0.996 − 0.0804i)26-s + (0.470 + 0.297i)29-s + (0.948 − 0.316i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.976 - 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5414964622\)
\(L(\frac12)\) \(\approx\) \(0.5414964622\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.845 + 0.534i)T \)
5 \( 1 + (0.948 + 0.316i)T \)
13 \( 1 + (0.799 - 0.600i)T \)
good3 \( 1 + (-0.0402 + 0.999i)T^{2} \)
7 \( 1 + (-0.278 - 0.960i)T^{2} \)
11 \( 1 + (-0.996 + 0.0804i)T^{2} \)
17 \( 1 + (1.08 - 1.44i)T + (-0.278 - 0.960i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.470 - 0.297i)T + (0.428 + 0.903i)T^{2} \)
31 \( 1 + (0.120 - 0.992i)T^{2} \)
37 \( 1 + (-1.87 - 0.625i)T + (0.799 + 0.600i)T^{2} \)
41 \( 1 + (-0.770 + 0.740i)T + (0.0402 - 0.999i)T^{2} \)
43 \( 1 + (0.799 - 0.600i)T^{2} \)
47 \( 1 + (0.354 - 0.935i)T^{2} \)
53 \( 1 + (1.19 + 0.144i)T + (0.970 + 0.239i)T^{2} \)
59 \( 1 + (-0.200 + 0.979i)T^{2} \)
61 \( 1 + (-1.27 - 0.543i)T + (0.692 + 0.721i)T^{2} \)
67 \( 1 + (0.632 + 0.774i)T^{2} \)
71 \( 1 + (-0.845 - 0.534i)T^{2} \)
73 \( 1 + (0.885 + 0.464i)T + (0.568 + 0.822i)T^{2} \)
79 \( 1 + (0.354 - 0.935i)T^{2} \)
83 \( 1 + (-0.885 - 0.464i)T^{2} \)
89 \( 1 + (-1.61 + 0.935i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.47 - 0.240i)T + (0.948 + 0.316i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.982679676078575520704854679420, −8.169595260708951762817419973989, −7.55311024409242663528891106464, −6.73721124462312164469148383058, −6.14466203201158734314457086713, −4.56598207507940202590528838368, −4.10529352540760254862414665954, −3.23693553536998057272485813083, −2.18672790393833107486330783249, −0.957263438171235150905952331421, 0.54207409485137708502703971878, 2.27071267920749361226912750396, 2.88197238867950813583284881211, 4.46160812766347234409954047321, 4.90324070540418520001784263033, 5.90114011409240352926733970260, 6.88231952196781301970999929931, 7.39530449504129354002584107422, 7.948136418024563147239679049057, 8.552436742327178383210259860718

Graph of the $Z$-function along the critical line