Properties

Label 2-3380-3380.2859-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.828 - 0.560i$
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  + (0.568 − 0.822i)2-s + (−0.354 − 0.935i)4-s + (−0.120 + 0.992i)5-s + (−0.970 − 0.239i)8-s + (−0.568 + 0.822i)9-s + (0.748 + 0.663i)10-s + (0.970 − 0.239i)13-s + (−0.748 + 0.663i)16-s + (−0.447 + 1.81i)17-s + (0.354 + 0.935i)18-s + (0.970 − 0.239i)20-s + (−0.970 − 0.239i)25-s + (0.354 − 0.935i)26-s + (−1.00 + 1.45i)29-s + (0.120 + 0.992i)32-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (−0.354 − 0.935i)4-s + (−0.120 + 0.992i)5-s + (−0.970 − 0.239i)8-s + (−0.568 + 0.822i)9-s + (0.748 + 0.663i)10-s + (0.970 − 0.239i)13-s + (−0.748 + 0.663i)16-s + (−0.447 + 1.81i)17-s + (0.354 + 0.935i)18-s + (0.970 − 0.239i)20-s + (−0.970 − 0.239i)25-s + (0.354 − 0.935i)26-s + (−1.00 + 1.45i)29-s + (0.120 + 0.992i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.828 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.247480380\)
\(L(\frac12)\) \(\approx\) \(1.247480380\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.568 + 0.822i)T \)
5 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (-0.970 + 0.239i)T \)
good3 \( 1 + (0.568 - 0.822i)T^{2} \)
7 \( 1 + (-0.885 - 0.464i)T^{2} \)
11 \( 1 + (-0.354 + 0.935i)T^{2} \)
17 \( 1 + (0.447 - 1.81i)T + (-0.885 - 0.464i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (1.00 - 1.45i)T + (-0.354 - 0.935i)T^{2} \)
31 \( 1 + (-0.970 - 0.239i)T^{2} \)
37 \( 1 + (0.180 - 1.48i)T + (-0.970 - 0.239i)T^{2} \)
41 \( 1 + (0.764 + 1.45i)T + (-0.568 + 0.822i)T^{2} \)
43 \( 1 + (-0.970 + 0.239i)T^{2} \)
47 \( 1 + (0.748 + 0.663i)T^{2} \)
53 \( 1 + (-0.114 + 0.464i)T + (-0.885 - 0.464i)T^{2} \)
59 \( 1 + (0.120 + 0.992i)T^{2} \)
61 \( 1 + (-1.71 + 0.423i)T + (0.885 - 0.464i)T^{2} \)
67 \( 1 + (0.748 + 0.663i)T^{2} \)
71 \( 1 + (0.568 - 0.822i)T^{2} \)
73 \( 1 + (-1.13 - 1.64i)T + (-0.354 + 0.935i)T^{2} \)
79 \( 1 + (0.748 + 0.663i)T^{2} \)
83 \( 1 + (-0.568 - 0.822i)T^{2} \)
89 \( 1 + 1.32iT - T^{2} \)
97 \( 1 + (-0.180 + 0.159i)T + (0.120 - 0.992i)T^{2} \)
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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

−8.671761628755926364926100878992, −8.442720162707367486004704035644, −7.25021900264641311146029518016, −6.41018064821124004151177482336, −5.78695767289563452745425589359, −5.05538068349408712644075947057, −3.83715357888601553130770947562, −3.48127067683739416500989693439, −2.40732949623937286026294246629, −1.62069682164278821922300952750, 0.60449211509219594682825792495, 2.37033550112154686413848818541, 3.52685860719350768839987813033, 4.14302979831630649685704993414, 5.02405820467387910073980359412, 5.69081191151060383197395229130, 6.36120271189793943934320107738, 7.19190862463913193104873312701, 7.950575407567353272138812314963, 8.703134236691893232671205896107

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