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Properties

Label 2-59e2-59.56-c0-0-2
Degree $2$
Conductor $3481$
Sign $0.914 - 0.405i$
Analytic cond. $1.73724$
Root an. cond. $1.31804$
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  + (0.947 + 0.319i)3-s + (0.647 − 0.762i)4-s + (0.856 + 0.515i)5-s + (−0.0541 + 0.998i)7-s + (0.856 − 0.515i)12-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.108 + 1.99i)17-s + (0.725 − 0.687i)19-s + (0.947 − 0.319i)20-s + (−0.370 + 0.928i)21-s + (−0.561 − 0.827i)27-s + (0.725 + 0.687i)28-s + (−0.907 − 0.419i)29-s + (−0.561 + 0.827i)35-s + ⋯
L(s)  = 1  + (0.947 + 0.319i)3-s + (0.647 − 0.762i)4-s + (0.856 + 0.515i)5-s + (−0.0541 + 0.998i)7-s + (0.856 − 0.515i)12-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.108 + 1.99i)17-s + (0.725 − 0.687i)19-s + (0.947 − 0.319i)20-s + (−0.370 + 0.928i)21-s + (−0.561 − 0.827i)27-s + (0.725 + 0.687i)28-s + (−0.907 − 0.419i)29-s + (−0.561 + 0.827i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3481\)    =    \(59^{2}\)
Sign: $0.914 - 0.405i$
Analytic conductor: \(1.73724\)
Root analytic conductor: \(1.31804\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3481} (3183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3481,\ (\ :0),\ 0.914 - 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.294918038\)
\(L(\frac12)\) \(\approx\) \(2.294918038\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.647 + 0.762i)T^{2} \)
3 \( 1 + (-0.947 - 0.319i)T + (0.796 + 0.605i)T^{2} \)
5 \( 1 + (-0.856 - 0.515i)T + (0.468 + 0.883i)T^{2} \)
7 \( 1 + (0.0541 - 0.998i)T + (-0.994 - 0.108i)T^{2} \)
11 \( 1 + (0.947 + 0.319i)T^{2} \)
13 \( 1 + (-0.267 + 0.963i)T^{2} \)
17 \( 1 + (-0.108 - 1.99i)T + (-0.994 + 0.108i)T^{2} \)
19 \( 1 + (-0.725 + 0.687i)T + (0.0541 - 0.998i)T^{2} \)
23 \( 1 + (-0.907 + 0.419i)T^{2} \)
29 \( 1 + (0.907 + 0.419i)T + (0.647 + 0.762i)T^{2} \)
31 \( 1 + (-0.0541 - 0.998i)T^{2} \)
37 \( 1 + (0.856 - 0.515i)T^{2} \)
41 \( 1 + (0.976 + 0.214i)T + (0.907 + 0.419i)T^{2} \)
43 \( 1 + (0.947 - 0.319i)T^{2} \)
47 \( 1 + (-0.468 + 0.883i)T^{2} \)
53 \( 1 + (-0.994 + 0.108i)T + (0.976 - 0.214i)T^{2} \)
61 \( 1 + (-0.647 + 0.762i)T^{2} \)
67 \( 1 + (0.856 + 0.515i)T^{2} \)
71 \( 1 + (1.71 - 1.03i)T + (0.468 - 0.883i)T^{2} \)
73 \( 1 + (0.725 - 0.687i)T^{2} \)
79 \( 1 + (-0.947 + 0.319i)T + (0.796 - 0.605i)T^{2} \)
83 \( 1 + (0.370 + 0.928i)T^{2} \)
89 \( 1 + (-0.647 - 0.762i)T^{2} \)
97 \( 1 + (0.725 + 0.687i)T^{2} \)
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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

−8.904002686746150143039648940193, −8.267103434965461917494658332061, −7.30035994352926043356001965809, −6.36723743422722790026311991955, −5.88906537051171580616328696002, −5.32445119537300393111100181410, −3.98298000374475051641724776416, −3.01518361368731930635411564877, −2.34220998619901100547215421277, −1.69082957410604514727766461662, 1.35272216614411083307350701628, 2.26437484839307484112448612627, 3.10432100615457618352328521119, 3.74767357878982172209863838179, 4.93690550792149623226647539735, 5.69393984384332756783128635599, 6.80381259871860466973075404473, 7.44574261141511593352004592531, 7.74666810177202173205258924339, 8.738458391747015765754317593733

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