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Properties

Label 2-3380-3380.2607-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.998 + 0.0598i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
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  + (0.903 − 0.428i)2-s + (0.632 − 0.774i)4-s + (0.692 + 0.721i)5-s + (0.239 − 0.970i)8-s + (−0.0804 + 0.996i)9-s + (0.935 + 0.354i)10-s + (−0.278 + 0.960i)13-s + (−0.200 − 0.979i)16-s + (−0.176 − 0.0969i)17-s + (0.354 + 0.935i)18-s + (0.996 − 0.0804i)20-s + (−0.0402 + 0.999i)25-s + (0.160 + 0.987i)26-s + (1.52 − 0.724i)29-s + (−0.600 − 0.799i)32-s + ⋯
L(s)  = 1  + (0.903 − 0.428i)2-s + (0.632 − 0.774i)4-s + (0.692 + 0.721i)5-s + (0.239 − 0.970i)8-s + (−0.0804 + 0.996i)9-s + (0.935 + 0.354i)10-s + (−0.278 + 0.960i)13-s + (−0.200 − 0.979i)16-s + (−0.176 − 0.0969i)17-s + (0.354 + 0.935i)18-s + (0.996 − 0.0804i)20-s + (−0.0402 + 0.999i)25-s + (0.160 + 0.987i)26-s + (1.52 − 0.724i)29-s + (−0.600 − 0.799i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.998 + 0.0598i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.998 + 0.0598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.468688157\)
\(L(\frac12)\) \(\approx\) \(2.468688157\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.903 + 0.428i)T \)
5 \( 1 + (-0.692 - 0.721i)T \)
13 \( 1 + (0.278 - 0.960i)T \)
good3 \( 1 + (0.0804 - 0.996i)T^{2} \)
7 \( 1 + (0.845 - 0.534i)T^{2} \)
11 \( 1 + (0.160 + 0.987i)T^{2} \)
17 \( 1 + (0.176 + 0.0969i)T + (0.534 + 0.845i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-1.52 + 0.724i)T + (0.632 - 0.774i)T^{2} \)
31 \( 1 + (-0.239 + 0.970i)T^{2} \)
37 \( 1 + (-0.506 - 0.380i)T + (0.278 + 0.960i)T^{2} \)
41 \( 1 + (-1.38 + 1.27i)T + (0.0804 - 0.996i)T^{2} \)
43 \( 1 + (-0.960 - 0.278i)T^{2} \)
47 \( 1 + (0.748 + 0.663i)T^{2} \)
53 \( 1 + (0.825 - 0.499i)T + (0.464 - 0.885i)T^{2} \)
59 \( 1 + (-0.391 + 0.919i)T^{2} \)
61 \( 1 + (1.38 + 1.44i)T + (-0.0402 + 0.999i)T^{2} \)
67 \( 1 + (-0.200 + 0.979i)T^{2} \)
71 \( 1 + (0.903 - 0.428i)T^{2} \)
73 \( 1 + (0.822 - 0.568i)T + (0.354 - 0.935i)T^{2} \)
79 \( 1 + (-0.748 - 0.663i)T^{2} \)
83 \( 1 + (-0.568 - 0.822i)T^{2} \)
89 \( 1 + (1.92 - 0.516i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.628 + 1.88i)T + (-0.799 - 0.600i)T^{2} \)
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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

−8.976522683302745381414792757321, −7.81333289126784137970893210905, −7.10597335771036114395258723514, −6.35233166212696057334351536172, −5.79709577197059398577045715831, −4.81395154256590254696239841119, −4.29877262869242789485202923034, −3.04672685704762178504521973879, −2.41493731781200731506890920457, −1.61340399972871653523371422267, 1.22175820384110944478128283519, 2.58165295873418800491195823970, 3.28936771675559231388682190185, 4.40384092068913056899126077970, 4.94918250757423534838949596199, 5.91651958780361044021295072179, 6.23707450903177296309758478383, 7.15670474886417504101142902522, 8.046857794417080164976848795107, 8.654951067993459604627185380839

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