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Properties

Label 2-3479-497.137-c0-0-0
Degree $2$
Conductor $3479$
Sign $0.930 + 0.367i$
Analytic cond. $1.73624$
Root an. cond. $1.31766$
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  + (0.413 − 0.459i)2-s + (0.0646 + 0.614i)4-s + (0.809 + 0.587i)8-s + (−0.913 − 0.406i)9-s + (0.244 − 1.14i)11-s + (−0.564 + 0.251i)18-s + (−0.427 − 0.587i)22-s + (1.64 − 0.951i)23-s + (0.978 + 0.207i)25-s + (1.61 + 1.17i)29-s + (−0.5 + 0.866i)32-s + (0.190 − 0.587i)36-s + (−0.809 − 1.40i)37-s + (−0.5 − 0.363i)43-s + (0.722 + 0.0759i)44-s + ⋯
L(s)  = 1  + (0.413 − 0.459i)2-s + (0.0646 + 0.614i)4-s + (0.809 + 0.587i)8-s + (−0.913 − 0.406i)9-s + (0.244 − 1.14i)11-s + (−0.564 + 0.251i)18-s + (−0.427 − 0.587i)22-s + (1.64 − 0.951i)23-s + (0.978 + 0.207i)25-s + (1.61 + 1.17i)29-s + (−0.5 + 0.866i)32-s + (0.190 − 0.587i)36-s + (−0.809 − 1.40i)37-s + (−0.5 − 0.363i)43-s + (0.722 + 0.0759i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3479\)    =    \(7^{2} \cdot 71\)
Sign: $0.930 + 0.367i$
Analytic conductor: \(1.73624\)
Root analytic conductor: \(1.31766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3479} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3479,\ (\ :0),\ 0.930 + 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.656363446\)
\(L(\frac12)\) \(\approx\) \(1.656363446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.413 + 0.459i)T + (-0.104 - 0.994i)T^{2} \)
3 \( 1 + (0.913 + 0.406i)T^{2} \)
5 \( 1 + (-0.978 - 0.207i)T^{2} \)
11 \( 1 + (-0.244 + 1.14i)T + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.104 - 0.994i)T^{2} \)
19 \( 1 + (-0.104 - 0.994i)T^{2} \)
23 \( 1 + (-1.64 + 0.951i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.913 - 0.406i)T^{2} \)
37 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (-0.913 + 0.406i)T^{2} \)
53 \( 1 + (-0.773 - 1.73i)T + (-0.669 + 0.743i)T^{2} \)
59 \( 1 + (-0.669 + 0.743i)T^{2} \)
61 \( 1 + (-0.913 + 0.406i)T^{2} \)
67 \( 1 + (-1.16 + 0.122i)T + (0.978 - 0.207i)T^{2} \)
73 \( 1 + (0.913 + 0.406i)T^{2} \)
79 \( 1 + (0.169 - 1.60i)T + (-0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.978 - 0.207i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.762067789601940777773636882276, −8.220937544400382742537794351076, −7.09234748990601879525542124328, −6.59434496985728474550755632333, −5.52454480236996159917278919624, −4.88494849002369750618071487401, −3.86688735155783132034128072482, −3.05668858966157572810019783807, −2.67174611248522110430302183238, −1.06444715209630226857682830461, 1.19146696513874801047739921982, 2.33103746689608100625082301616, 3.32436045559811134921303139562, 4.62649578083481095536487835803, 4.93862614229663641611484551304, 5.73787155027158122742396266689, 6.71503160581881413401556976983, 6.97762641427083541954983607195, 8.026911890257600990245302612130, 8.725123666981595838102878665135

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