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Properties

Label 2-59e2-59.40-c0-0-2
Degree $2$
Conductor $3481$
Sign $0.999 + 0.0251i$
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  + (0.725 − 0.687i)3-s + (−0.561 − 0.827i)4-s + (−0.976 + 0.214i)5-s + (−0.796 + 0.605i)7-s + (−0.976 − 0.214i)12-s + (−0.561 + 0.827i)15-s + (−0.370 + 0.928i)16-s + (1.59 + 1.21i)17-s + (0.947 − 0.319i)19-s + (0.725 + 0.687i)20-s + (−0.161 + 0.986i)21-s + (0.647 + 0.762i)27-s + (0.947 + 0.319i)28-s + (−0.468 − 0.883i)29-s + (0.647 − 0.762i)35-s + ⋯
L(s)  = 1  + (0.725 − 0.687i)3-s + (−0.561 − 0.827i)4-s + (−0.976 + 0.214i)5-s + (−0.796 + 0.605i)7-s + (−0.976 − 0.214i)12-s + (−0.561 + 0.827i)15-s + (−0.370 + 0.928i)16-s + (1.59 + 1.21i)17-s + (0.947 − 0.319i)19-s + (0.725 + 0.687i)20-s + (−0.161 + 0.986i)21-s + (0.647 + 0.762i)27-s + (0.947 + 0.319i)28-s + (−0.468 − 0.883i)29-s + (0.647 − 0.762i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3481\)    =    \(59^{2}\)
Sign: $0.999 + 0.0251i$
Analytic conductor: \(1.73724\)
Root analytic conductor: \(1.31804\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3481} (3344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3481,\ (\ :0),\ 0.999 + 0.0251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.040006178\)
\(L(\frac12)\) \(\approx\) \(1.040006178\)
\(L(1)\) 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 \( 1 + (0.561 + 0.827i)T^{2} \)
3 \( 1 + (-0.725 + 0.687i)T + (0.0541 - 0.998i)T^{2} \)
5 \( 1 + (0.976 - 0.214i)T + (0.907 - 0.419i)T^{2} \)
7 \( 1 + (0.796 - 0.605i)T + (0.267 - 0.963i)T^{2} \)
11 \( 1 + (0.725 - 0.687i)T^{2} \)
13 \( 1 + (0.994 - 0.108i)T^{2} \)
17 \( 1 + (-1.59 - 1.21i)T + (0.267 + 0.963i)T^{2} \)
19 \( 1 + (-0.947 + 0.319i)T + (0.796 - 0.605i)T^{2} \)
23 \( 1 + (-0.468 + 0.883i)T^{2} \)
29 \( 1 + (0.468 + 0.883i)T + (-0.561 + 0.827i)T^{2} \)
31 \( 1 + (-0.796 - 0.605i)T^{2} \)
37 \( 1 + (-0.976 - 0.214i)T^{2} \)
41 \( 1 + (-0.856 - 0.515i)T + (0.468 + 0.883i)T^{2} \)
43 \( 1 + (0.725 + 0.687i)T^{2} \)
47 \( 1 + (-0.907 - 0.419i)T^{2} \)
53 \( 1 + (0.267 + 0.963i)T + (-0.856 + 0.515i)T^{2} \)
61 \( 1 + (0.561 + 0.827i)T^{2} \)
67 \( 1 + (-0.976 + 0.214i)T^{2} \)
71 \( 1 + (-1.95 - 0.429i)T + (0.907 + 0.419i)T^{2} \)
73 \( 1 + (0.947 - 0.319i)T^{2} \)
79 \( 1 + (-0.725 - 0.687i)T + (0.0541 + 0.998i)T^{2} \)
83 \( 1 + (0.161 + 0.986i)T^{2} \)
89 \( 1 + (0.561 - 0.827i)T^{2} \)
97 \( 1 + (0.947 + 0.319i)T^{2} \)
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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

−8.631007650258550843033468974483, −7.968728743021393553492068510376, −7.53864248765419460599599989832, −6.52066076161392886668413573622, −5.80565224018136763385072686729, −5.07104199227678535460145674600, −3.87723432005054697390054803239, −3.29278322512744774366850257742, −2.24285364860173734350236049872, −1.08656361323086503657144658857, 0.71098570169682273089090606131, 2.89235829391290085178319652223, 3.44275367727035502787734385019, 3.82695227544426423873497208894, 4.66628814920803294753616698879, 5.52963099186648182450320480280, 6.86410401452832840767674846085, 7.58996316339739208140018877080, 7.929474076413019127110533858513, 8.827870924961847725768195150139

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