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Properties

Label 2-3380-3380.3319-c0-0-0
Degree $2$
Conductor $3380$
Sign $-0.394 + 0.918i$
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  + (−0.948 + 0.316i)2-s + (0.799 − 0.600i)4-s + (0.428 − 0.903i)5-s + (−0.568 + 0.822i)8-s + (0.200 − 0.979i)9-s + (−0.120 + 0.992i)10-s + (−0.996 + 0.0804i)13-s + (0.278 − 0.960i)16-s + (0.0966 − 1.19i)17-s + (0.120 + 0.992i)18-s + (−0.200 − 0.979i)20-s + (−0.632 − 0.774i)25-s + (0.919 − 0.391i)26-s + (−1.87 + 0.625i)29-s + (0.0402 + 0.999i)32-s + ⋯
L(s)  = 1  + (−0.948 + 0.316i)2-s + (0.799 − 0.600i)4-s + (0.428 − 0.903i)5-s + (−0.568 + 0.822i)8-s + (0.200 − 0.979i)9-s + (−0.120 + 0.992i)10-s + (−0.996 + 0.0804i)13-s + (0.278 − 0.960i)16-s + (0.0966 − 1.19i)17-s + (0.120 + 0.992i)18-s + (−0.200 − 0.979i)20-s + (−0.632 − 0.774i)25-s + (0.919 − 0.391i)26-s + (−1.87 + 0.625i)29-s + (0.0402 + 0.999i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.394 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6415257773\)
\(L(\frac12)\) \(\approx\) \(0.6415257773\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.948 - 0.316i)T \)
5 \( 1 + (-0.428 + 0.903i)T \)
13 \( 1 + (0.996 - 0.0804i)T \)
good3 \( 1 + (-0.200 + 0.979i)T^{2} \)
7 \( 1 + (-0.987 - 0.160i)T^{2} \)
11 \( 1 + (-0.919 + 0.391i)T^{2} \)
17 \( 1 + (-0.0966 + 1.19i)T + (-0.987 - 0.160i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.87 - 0.625i)T + (0.799 - 0.600i)T^{2} \)
31 \( 1 + (0.568 - 0.822i)T^{2} \)
37 \( 1 + (-0.0557 + 1.38i)T + (-0.996 - 0.0804i)T^{2} \)
41 \( 1 + (0.490 - 0.400i)T + (0.200 - 0.979i)T^{2} \)
43 \( 1 + (-0.996 + 0.0804i)T^{2} \)
47 \( 1 + (0.970 + 0.239i)T^{2} \)
53 \( 1 + (0.132 + 0.0914i)T + (0.354 + 0.935i)T^{2} \)
59 \( 1 + (-0.845 + 0.534i)T^{2} \)
61 \( 1 + (-0.542 + 1.14i)T + (-0.632 - 0.774i)T^{2} \)
67 \( 1 + (-0.278 - 0.960i)T^{2} \)
71 \( 1 + (0.948 - 0.316i)T^{2} \)
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 \( 1 + (-1.22 - 1.27i)T + (-0.0402 + 0.999i)T^{2} \)
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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

−8.814574033470315172810285275733, −7.76308517718474924287501483297, −7.23195079855714587471178598562, −6.47472255046340300921438654873, −5.53663219922297708015135768568, −5.06603509530472327272437942983, −3.88280101220937393105966217914, −2.64006395522564138006317404996, −1.67373907642454613357573739850, −0.49100637333742847054420690442, 1.72265247423201932572210921674, 2.31105772093201506536088695898, 3.24305469424099674457482788609, 4.21022990081801128484800505803, 5.45455642854343274080618992198, 6.17400679538180789747738090324, 7.11542160281966466488977641862, 7.53750200199943505214290882750, 8.242529055005619105838274873517, 9.099508309598642707672571374789

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