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Properties

Label 2-59e2-59.13-c0-0-0
Degree $2$
Conductor $3481$
Sign $-0.572 + 0.819i$
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  + (−0.796 + 0.605i)3-s + (−0.161 + 0.986i)4-s + (−0.468 + 0.883i)5-s + (0.994 − 0.108i)7-s + (−0.468 − 0.883i)12-s + (−0.161 − 0.986i)15-s + (−0.947 − 0.319i)16-s + (−1.98 − 0.216i)17-s + (−0.0541 − 0.998i)19-s + (−0.796 − 0.605i)20-s + (−0.725 + 0.687i)21-s + (−0.370 − 0.928i)27-s + (−0.0541 + 0.998i)28-s + (−0.647 + 0.762i)29-s + (−0.370 + 0.928i)35-s + ⋯
L(s)  = 1  + (−0.796 + 0.605i)3-s + (−0.161 + 0.986i)4-s + (−0.468 + 0.883i)5-s + (0.994 − 0.108i)7-s + (−0.468 − 0.883i)12-s + (−0.161 − 0.986i)15-s + (−0.947 − 0.319i)16-s + (−1.98 − 0.216i)17-s + (−0.0541 − 0.998i)19-s + (−0.796 − 0.605i)20-s + (−0.725 + 0.687i)21-s + (−0.370 − 0.928i)27-s + (−0.0541 + 0.998i)28-s + (−0.647 + 0.762i)29-s + (−0.370 + 0.928i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3481\)    =    \(59^{2}\)
Sign: $-0.572 + 0.819i$
Analytic conductor: \(1.73724\)
Root analytic conductor: \(1.31804\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3481} (1311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3481,\ (\ :0),\ -0.572 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3224696865\)
\(L(\frac12)\) \(\approx\) \(0.3224696865\)
\(L(1)\) 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 \( 1 + (0.161 - 0.986i)T^{2} \)
3 \( 1 + (0.796 - 0.605i)T + (0.267 - 0.963i)T^{2} \)
5 \( 1 + (0.468 - 0.883i)T + (-0.561 - 0.827i)T^{2} \)
7 \( 1 + (-0.994 + 0.108i)T + (0.976 - 0.214i)T^{2} \)
11 \( 1 + (-0.796 + 0.605i)T^{2} \)
13 \( 1 + (0.856 - 0.515i)T^{2} \)
17 \( 1 + (1.98 + 0.216i)T + (0.976 + 0.214i)T^{2} \)
19 \( 1 + (0.0541 + 0.998i)T + (-0.994 + 0.108i)T^{2} \)
23 \( 1 + (-0.647 - 0.762i)T^{2} \)
29 \( 1 + (0.647 - 0.762i)T + (-0.161 - 0.986i)T^{2} \)
31 \( 1 + (0.994 + 0.108i)T^{2} \)
37 \( 1 + (-0.468 - 0.883i)T^{2} \)
41 \( 1 + (0.907 - 0.419i)T + (0.647 - 0.762i)T^{2} \)
43 \( 1 + (-0.796 - 0.605i)T^{2} \)
47 \( 1 + (0.561 - 0.827i)T^{2} \)
53 \( 1 + (0.976 + 0.214i)T + (0.907 + 0.419i)T^{2} \)
61 \( 1 + (0.161 - 0.986i)T^{2} \)
67 \( 1 + (-0.468 + 0.883i)T^{2} \)
71 \( 1 + (-0.936 - 1.76i)T + (-0.561 + 0.827i)T^{2} \)
73 \( 1 + (-0.0541 - 0.998i)T^{2} \)
79 \( 1 + (0.796 + 0.605i)T + (0.267 + 0.963i)T^{2} \)
83 \( 1 + (0.725 + 0.687i)T^{2} \)
89 \( 1 + (0.161 + 0.986i)T^{2} \)
97 \( 1 + (-0.0541 + 0.998i)T^{2} \)
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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

−9.065592235349095788368994467012, −8.513315042106715431666959285089, −7.70026644715467537848562149659, −7.03412199850922738883510912422, −6.47041704142967023671452854386, −5.15487541891723665812063379991, −4.66553803671511445184329849121, −4.01368957258231753078889585435, −3.00807103149647199034465115964, −2.10299183705633842208068385343, 0.20943424091752786736050272788, 1.41077548446022280969860339567, 2.07124994796079228364226414875, 3.91942254964707014733297486553, 4.69211157296652603576319705694, 5.19234888162753563989825319743, 6.07399124276222268137595971189, 6.54965790627511418612335932938, 7.52481402352806569200480018617, 8.388798842870355551607475026011

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