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Properties

Label 2-3380-3380.3059-c0-0-1
Degree $2$
Conductor $3380$
Sign $-0.864 + 0.502i$
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  + (−0.200 − 0.979i)2-s + (−0.919 + 0.391i)4-s + (0.845 − 0.534i)5-s + (0.568 + 0.822i)8-s + (−0.948 − 0.316i)9-s + (−0.692 − 0.721i)10-s + (−0.428 − 0.903i)13-s + (0.692 − 0.721i)16-s + (0.708 + 0.336i)17-s + (−0.120 + 0.992i)18-s + (−0.568 + 0.822i)20-s + (0.428 − 0.903i)25-s + (−0.799 + 0.600i)26-s + (−0.253 − 1.23i)29-s + (−0.845 − 0.534i)32-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.919 + 0.391i)4-s + (0.845 − 0.534i)5-s + (0.568 + 0.822i)8-s + (−0.948 − 0.316i)9-s + (−0.692 − 0.721i)10-s + (−0.428 − 0.903i)13-s + (0.692 − 0.721i)16-s + (0.708 + 0.336i)17-s + (−0.120 + 0.992i)18-s + (−0.568 + 0.822i)20-s + (0.428 − 0.903i)25-s + (−0.799 + 0.600i)26-s + (−0.253 − 1.23i)29-s + (−0.845 − 0.534i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.864 + 0.502i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.864 + 0.502i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9613829517\)
\(L(\frac12)\) \(\approx\) \(0.9613829517\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.200 + 0.979i)T \)
5 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (0.428 + 0.903i)T \)
good3 \( 1 + (0.948 + 0.316i)T^{2} \)
7 \( 1 + (0.632 + 0.774i)T^{2} \)
11 \( 1 + (0.799 - 0.600i)T^{2} \)
17 \( 1 + (-0.708 - 0.336i)T + (0.632 + 0.774i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.253 + 1.23i)T + (-0.919 + 0.391i)T^{2} \)
31 \( 1 + (0.568 + 0.822i)T^{2} \)
37 \( 1 + (0.470 - 0.297i)T + (0.428 - 0.903i)T^{2} \)
41 \( 1 + (-0.314 + 1.93i)T + (-0.948 - 0.316i)T^{2} \)
43 \( 1 + (0.428 + 0.903i)T^{2} \)
47 \( 1 + (0.970 - 0.239i)T^{2} \)
53 \( 1 + (1.48 - 1.02i)T + (0.354 - 0.935i)T^{2} \)
59 \( 1 + (-0.0402 + 0.999i)T^{2} \)
61 \( 1 + (-1.96 - 0.158i)T + (0.987 + 0.160i)T^{2} \)
67 \( 1 + (-0.692 - 0.721i)T^{2} \)
71 \( 1 + (-0.200 - 0.979i)T^{2} \)
73 \( 1 + (-0.748 - 0.663i)T + (0.120 + 0.992i)T^{2} \)
79 \( 1 + (0.970 - 0.239i)T^{2} \)
83 \( 1 + (0.748 + 0.663i)T^{2} \)
89 \( 1 + (0.414 + 0.239i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.492 + 1.70i)T + (-0.845 + 0.534i)T^{2} \)
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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

−8.416674896639948087427121854887, −8.265073458398625454603155186699, −7.11378214461293361054560940223, −5.76605294111865616037638256404, −5.56898221762340262593917299109, −4.57931585440466241391906709658, −3.55192524903830726563375400248, −2.73881537226197686670221963168, −1.88929588942929385043000750927, −0.62163294564325618631008490103, 1.51266932167668045732360558105, 2.70391850703256293904530577291, 3.67409800277103436103388581945, 5.01185585575051487180646506010, 5.28226774741705745335685033627, 6.34464382835970342123591898177, 6.67048358883350338776461281001, 7.62390720935789994237149332545, 8.223738785464485541253324697467, 9.247332351821556154278819084846

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