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Properties

Label 2-3380-3380.2799-c0-0-1
Degree $2$
Conductor $3380$
Sign $0.817 - 0.575i$
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  + (0.996 − 0.0804i)2-s + (0.987 − 0.160i)4-s + (0.278 + 0.960i)5-s + (0.970 − 0.239i)8-s + (−0.428 + 0.903i)9-s + (0.354 + 0.935i)10-s + (0.692 − 0.721i)13-s + (0.948 − 0.316i)16-s + (−0.231 + 0.222i)17-s + (−0.354 + 0.935i)18-s + (0.428 + 0.903i)20-s + (−0.845 + 0.534i)25-s + (0.632 − 0.774i)26-s + (−0.0802 + 0.00648i)29-s + (0.919 − 0.391i)32-s + ⋯
L(s)  = 1  + (0.996 − 0.0804i)2-s + (0.987 − 0.160i)4-s + (0.278 + 0.960i)5-s + (0.970 − 0.239i)8-s + (−0.428 + 0.903i)9-s + (0.354 + 0.935i)10-s + (0.692 − 0.721i)13-s + (0.948 − 0.316i)16-s + (−0.231 + 0.222i)17-s + (−0.354 + 0.935i)18-s + (0.428 + 0.903i)20-s + (−0.845 + 0.534i)25-s + (0.632 − 0.774i)26-s + (−0.0802 + 0.00648i)29-s + (0.919 − 0.391i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.817 - 0.575i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.817 - 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.563438327\)
\(L(\frac12)\) \(\approx\) \(2.563438327\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 + 0.0804i)T \)
5 \( 1 + (-0.278 - 0.960i)T \)
13 \( 1 + (-0.692 + 0.721i)T \)
good3 \( 1 + (0.428 - 0.903i)T^{2} \)
7 \( 1 + (0.0402 - 0.999i)T^{2} \)
11 \( 1 + (-0.632 + 0.774i)T^{2} \)
17 \( 1 + (0.231 - 0.222i)T + (0.0402 - 0.999i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.0802 - 0.00648i)T + (0.987 - 0.160i)T^{2} \)
31 \( 1 + (-0.970 + 0.239i)T^{2} \)
37 \( 1 + (0.368 + 0.156i)T + (0.692 + 0.721i)T^{2} \)
41 \( 1 + (0.0860 + 0.136i)T + (-0.428 + 0.903i)T^{2} \)
43 \( 1 + (0.692 - 0.721i)T^{2} \)
47 \( 1 + (0.748 - 0.663i)T^{2} \)
53 \( 1 + (0.345 + 1.40i)T + (-0.885 + 0.464i)T^{2} \)
59 \( 1 + (0.799 + 0.600i)T^{2} \)
61 \( 1 + (-0.470 - 1.62i)T + (-0.845 + 0.534i)T^{2} \)
67 \( 1 + (-0.948 - 0.316i)T^{2} \)
71 \( 1 + (-0.996 + 0.0804i)T^{2} \)
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 \( 1 + (1.14 + 0.663i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.0482 - 0.236i)T + (-0.919 - 0.391i)T^{2} \)
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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

−8.708626770500751868917196602311, −7.897951939754577799461432507489, −7.25906326212794796114637896460, −6.44253237445349270438829087064, −5.78951210436594426259396209927, −5.19361896906308238984322834012, −4.15654493346372559179283537287, −3.28282683699367450775879887068, −2.60915913836760723918008600785, −1.69664401508335980086262831829, 1.21766885343649499296482216519, 2.26256197421792664234851454231, 3.41584717058310771229560057127, 4.09956770095104627697220315734, 4.90173923253851269220506312186, 5.65813541600204255864544023236, 6.33536250185324519116028709139, 6.93319069627565161989153925757, 8.005647659311027327317663533382, 8.689294704525399168302353425598

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