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Properties

Label 2-3380-3380.3283-c0-0-0
Degree $2$
Conductor $3380$
Sign $-0.844 + 0.535i$
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  + (−0.428 − 0.903i)2-s + (−0.632 + 0.774i)4-s + (−0.600 + 0.799i)5-s + (0.970 + 0.239i)8-s + (0.0804 − 0.996i)9-s + (0.979 + 0.200i)10-s + (−0.960 − 0.278i)13-s + (−0.200 − 0.979i)16-s + (−0.0969 + 0.176i)17-s + (−0.935 + 0.354i)18-s + (−0.239 − 0.970i)20-s + (−0.278 − 0.960i)25-s + (0.160 + 0.987i)26-s + (−1.52 + 0.724i)29-s + (−0.799 + 0.600i)32-s + ⋯
L(s)  = 1  + (−0.428 − 0.903i)2-s + (−0.632 + 0.774i)4-s + (−0.600 + 0.799i)5-s + (0.970 + 0.239i)8-s + (0.0804 − 0.996i)9-s + (0.979 + 0.200i)10-s + (−0.960 − 0.278i)13-s + (−0.200 − 0.979i)16-s + (−0.0969 + 0.176i)17-s + (−0.935 + 0.354i)18-s + (−0.239 − 0.970i)20-s + (−0.278 − 0.960i)25-s + (0.160 + 0.987i)26-s + (−1.52 + 0.724i)29-s + (−0.799 + 0.600i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.844 + 0.535i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4608318597\)
\(L(\frac12)\) \(\approx\) \(0.4608318597\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.428 + 0.903i)T \)
5 \( 1 + (0.600 - 0.799i)T \)
13 \( 1 + (0.960 + 0.278i)T \)
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 \( 1 + (0.0969 - 0.176i)T + (-0.534 - 0.845i)T^{2} \)
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 \( 1 + (-0.380 + 0.506i)T + (-0.278 - 0.960i)T^{2} \)
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 \( 1 + (0.499 + 0.825i)T + (-0.464 + 0.885i)T^{2} \)
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 \( 1 + (0.568 + 0.822i)T + (-0.354 + 0.935i)T^{2} \)
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 \( 1 + (1.88 + 0.628i)T + (0.799 + 0.600i)T^{2} \)
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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

−8.686567109253104803947982392849, −7.59694481804202008475412910825, −7.37726277951213890682632534255, −6.39574252694554115897886797215, −5.33643068265553870690588843038, −4.24640739070499063992052018639, −3.62436500907526897162014111955, −2.88858173193051722121654734550, −1.92389203570153696210344955610, −0.33545388653252017222579554630, 1.30098289358133063570760678458, 2.51137830758379606852532547926, 4.11412910027831810099025224630, 4.60500343932380245668726949089, 5.34170424986917352859907146585, 6.06144460800295236300789489225, 7.18060340960241569559352399953, 7.71008347966942600875269955633, 8.077514479187956893667216850685, 9.155750004784543702246440680255

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