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Properties

Label 2-3380-3380.2799-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.0711 - 0.997i$
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 + (0.919 + 0.391i)5-s + (−0.970 + 0.239i)8-s + (−0.428 + 0.903i)9-s + (−0.948 − 0.316i)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 + (0.970 + 0.239i)20-s + (0.692 + 0.721i)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 + (0.919 + 0.391i)5-s + (−0.970 + 0.239i)8-s + (−0.428 + 0.903i)9-s + (−0.948 − 0.316i)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 + (0.970 + 0.239i)20-s + (0.692 + 0.721i)25-s + (0.632 − 0.774i)26-s + (−0.0802 + 0.00648i)29-s + (−0.919 + 0.391i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.0711 - 0.997i$
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: \((2,\ 3380,\ (\ :0),\ 0.0711 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8253451896\)
\(L(\frac12)\) \(\approx\) \(0.8253451896\)
\(L(1)\) 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 \( 1 + (-0.919 - 0.391i)T \)
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

−9.027239325709119275955739797776, −8.331179678993627707237118286071, −7.40321423068210232241987052498, −7.00665705251648917984739905802, −6.00016845990871765740728251692, −5.50827265025032550033288081880, −4.47218769870832836656196473587, −2.94691666529619278976387768990, −2.38570000976359458105422440946, −1.45970766139489789817032390141, 0.67129042919256390312493975142, 1.87064352099522060738192048043, 2.77705482090581198860283783100, 3.64728685368425543872676971919, 5.05184239467747491962968685429, 5.77431334908800322402934972924, 6.44890159002411894027757268858, 7.16062689555156190988343013563, 8.147827457873183400948777174354, 8.617722444227306004079550528946

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