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Properties

Label 2-59e2-59.2-c0-0-0
Degree $2$
Conductor $3481$
Sign $-0.906 - 0.423i$
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  + (−0.907 + 0.419i)3-s + (−0.994 − 0.108i)4-s + (0.947 + 0.319i)5-s + (0.561 + 0.827i)7-s + (0.947 − 0.319i)12-s + (−0.994 + 0.108i)15-s + (0.976 + 0.214i)16-s + (−1.12 + 1.65i)17-s + (−0.468 + 0.883i)19-s + (−0.907 − 0.419i)20-s + (−0.856 − 0.515i)21-s + (0.267 − 0.963i)27-s + (−0.468 − 0.883i)28-s + (−0.0541 − 0.998i)29-s + (0.267 + 0.963i)35-s + ⋯
L(s)  = 1  + (−0.907 + 0.419i)3-s + (−0.994 − 0.108i)4-s + (0.947 + 0.319i)5-s + (0.561 + 0.827i)7-s + (0.947 − 0.319i)12-s + (−0.994 + 0.108i)15-s + (0.976 + 0.214i)16-s + (−1.12 + 1.65i)17-s + (−0.468 + 0.883i)19-s + (−0.907 − 0.419i)20-s + (−0.856 − 0.515i)21-s + (0.267 − 0.963i)27-s + (−0.468 − 0.883i)28-s + (−0.0541 − 0.998i)29-s + (0.267 + 0.963i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3481\)    =    \(59^{2}\)
Sign: $-0.906 - 0.423i$
Analytic conductor: \(1.73724\)
Root analytic conductor: \(1.31804\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3481} (946, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3481,\ (\ :0),\ -0.906 - 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5872703278\)
\(L(\frac12)\) \(\approx\) \(0.5872703278\)
\(L(1)\) 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 \( 1 + (0.994 + 0.108i)T^{2} \)
3 \( 1 + (0.907 - 0.419i)T + (0.647 - 0.762i)T^{2} \)
5 \( 1 + (-0.947 - 0.319i)T + (0.796 + 0.605i)T^{2} \)
7 \( 1 + (-0.561 - 0.827i)T + (-0.370 + 0.928i)T^{2} \)
11 \( 1 + (-0.907 + 0.419i)T^{2} \)
13 \( 1 + (0.161 - 0.986i)T^{2} \)
17 \( 1 + (1.12 - 1.65i)T + (-0.370 - 0.928i)T^{2} \)
19 \( 1 + (0.468 - 0.883i)T + (-0.561 - 0.827i)T^{2} \)
23 \( 1 + (-0.0541 + 0.998i)T^{2} \)
29 \( 1 + (0.0541 + 0.998i)T + (-0.994 + 0.108i)T^{2} \)
31 \( 1 + (0.561 - 0.827i)T^{2} \)
37 \( 1 + (0.947 - 0.319i)T^{2} \)
41 \( 1 + (-0.725 - 0.687i)T + (0.0541 + 0.998i)T^{2} \)
43 \( 1 + (-0.907 - 0.419i)T^{2} \)
47 \( 1 + (-0.796 + 0.605i)T^{2} \)
53 \( 1 + (-0.370 - 0.928i)T + (-0.725 + 0.687i)T^{2} \)
61 \( 1 + (0.994 + 0.108i)T^{2} \)
67 \( 1 + (0.947 + 0.319i)T^{2} \)
71 \( 1 + (1.89 - 0.638i)T + (0.796 - 0.605i)T^{2} \)
73 \( 1 + (-0.468 + 0.883i)T^{2} \)
79 \( 1 + (0.907 + 0.419i)T + (0.647 + 0.762i)T^{2} \)
83 \( 1 + (0.856 - 0.515i)T^{2} \)
89 \( 1 + (0.994 - 0.108i)T^{2} \)
97 \( 1 + (-0.468 - 0.883i)T^{2} \)
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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

−9.035712783832804204926805796820, −8.495227301800524103890451739822, −7.84570505547744043032540187217, −6.33692257306720366069815925194, −5.93766166772841335997831112386, −5.49699091713936619113388710893, −4.52722605756179307612789501348, −4.03948622938636689971790978765, −2.51393853402309555295046172591, −1.63822657200977329616014210692, 0.41428707401496727314708227372, 1.43777215930780433360841036351, 2.74959595021909907646371802781, 4.05160608839946403945862597815, 4.88500051274965509792829743379, 5.23753249998905123018713601346, 6.12253313403236956851529900003, 6.97645527247265983724059365551, 7.49788000978653264235177936717, 8.752993227101576425854884181278

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