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Properties

Label 2-3380-3380.383-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.818 - 0.573i$
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  + (0.316 + 0.948i)2-s + (−0.799 + 0.600i)4-s + (0.428 − 0.903i)5-s + (−0.822 − 0.568i)8-s + (−0.979 − 0.200i)9-s + (0.992 + 0.120i)10-s + (0.996 − 0.0804i)13-s + (0.278 − 0.960i)16-s + (1.28 + 1.50i)17-s + (−0.120 − 0.992i)18-s + (0.200 + 0.979i)20-s + (−0.632 − 0.774i)25-s + (0.391 + 0.919i)26-s + (−0.625 − 1.87i)29-s + (0.999 − 0.0402i)32-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)2-s + (−0.799 + 0.600i)4-s + (0.428 − 0.903i)5-s + (−0.822 − 0.568i)8-s + (−0.979 − 0.200i)9-s + (0.992 + 0.120i)10-s + (0.996 − 0.0804i)13-s + (0.278 − 0.960i)16-s + (1.28 + 1.50i)17-s + (−0.120 − 0.992i)18-s + (0.200 + 0.979i)20-s + (−0.632 − 0.774i)25-s + (0.391 + 0.919i)26-s + (−0.625 − 1.87i)29-s + (0.999 − 0.0402i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.818 - 0.573i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.818 - 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.420301410\)
\(L(\frac12)\) \(\approx\) \(1.420301410\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.316 - 0.948i)T \)
5 \( 1 + (-0.428 + 0.903i)T \)
13 \( 1 + (-0.996 + 0.0804i)T \)
good3 \( 1 + (0.979 + 0.200i)T^{2} \)
7 \( 1 + (-0.987 - 0.160i)T^{2} \)
11 \( 1 + (0.391 + 0.919i)T^{2} \)
17 \( 1 + (-1.28 - 1.50i)T + (-0.160 + 0.987i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.625 + 1.87i)T + (-0.799 + 0.600i)T^{2} \)
31 \( 1 + (0.822 + 0.568i)T^{2} \)
37 \( 1 + (-0.0580 + 1.44i)T + (-0.996 - 0.0804i)T^{2} \)
41 \( 1 + (-1.78 - 0.179i)T + (0.979 + 0.200i)T^{2} \)
43 \( 1 + (-0.0804 - 0.996i)T^{2} \)
47 \( 1 + (0.970 + 0.239i)T^{2} \)
53 \( 1 + (-1.49 + 0.274i)T + (0.935 - 0.354i)T^{2} \)
59 \( 1 + (-0.534 - 0.845i)T^{2} \)
61 \( 1 + (0.664 - 1.39i)T + (-0.632 - 0.774i)T^{2} \)
67 \( 1 + (0.278 + 0.960i)T^{2} \)
71 \( 1 + (0.316 + 0.948i)T^{2} \)
73 \( 1 + (-0.663 - 0.748i)T + (-0.120 + 0.992i)T^{2} \)
79 \( 1 + (-0.970 - 0.239i)T^{2} \)
83 \( 1 + (0.748 - 0.663i)T^{2} \)
89 \( 1 + (-0.267 - 0.999i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.670 + 0.643i)T + (0.0402 - 0.999i)T^{2} \)
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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

−8.717550503802671265876487034549, −8.105890817115715791891659481479, −7.56790742653017248779721540850, −6.26372291721405183492315688677, −5.73277662279346476540879444279, −5.55815531920154732676483867057, −4.13782111858846726055455030178, −3.80496597957116759505926663948, −2.47169665925952121950307818258, −0.943251840937850096302143508298, 1.17244578880822093606922565113, 2.41410101397926886016974387259, 3.11273183798426328358523820920, 3.66540100225384444799281919250, 4.99604580051880279808347901541, 5.57423171102884303452538647193, 6.23806020771007030577987685460, 7.21324585374728842745521430017, 8.085422669194973100493695881582, 9.053740472708010566802146051375

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