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Properties

Label 2-3380-3380.1423-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.468 + 0.883i$
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  + (0.391 − 0.919i)2-s + (−0.692 − 0.721i)4-s + (0.987 − 0.160i)5-s + (−0.935 + 0.354i)8-s + (0.600 + 0.799i)9-s + (0.239 − 0.970i)10-s + (0.632 + 0.774i)13-s + (−0.0402 + 0.999i)16-s + (−0.0400 + 0.00404i)17-s + (0.970 − 0.239i)18-s + (−0.799 − 0.600i)20-s + (0.948 − 0.316i)25-s + (0.960 − 0.278i)26-s + (0.156 − 0.368i)29-s + (0.903 + 0.428i)32-s + ⋯
L(s)  = 1  + (0.391 − 0.919i)2-s + (−0.692 − 0.721i)4-s + (0.987 − 0.160i)5-s + (−0.935 + 0.354i)8-s + (0.600 + 0.799i)9-s + (0.239 − 0.970i)10-s + (0.632 + 0.774i)13-s + (−0.0402 + 0.999i)16-s + (−0.0400 + 0.00404i)17-s + (0.970 − 0.239i)18-s + (−0.799 − 0.600i)20-s + (0.948 − 0.316i)25-s + (0.960 − 0.278i)26-s + (0.156 − 0.368i)29-s + (0.903 + 0.428i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.468 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.821777634\)
\(L(\frac12)\) \(\approx\) \(1.821777634\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.391 + 0.919i)T \)
5 \( 1 + (-0.987 + 0.160i)T \)
13 \( 1 + (-0.632 - 0.774i)T \)
good3 \( 1 + (-0.600 - 0.799i)T^{2} \)
7 \( 1 + (0.200 + 0.979i)T^{2} \)
11 \( 1 + (0.960 - 0.278i)T^{2} \)
17 \( 1 + (0.0400 - 0.00404i)T + (0.979 - 0.200i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.156 + 0.368i)T + (-0.692 - 0.721i)T^{2} \)
31 \( 1 + (0.935 - 0.354i)T^{2} \)
37 \( 1 + (-0.458 - 0.965i)T + (-0.632 + 0.774i)T^{2} \)
41 \( 1 + (-0.333 + 0.668i)T + (-0.600 - 0.799i)T^{2} \)
43 \( 1 + (0.774 - 0.632i)T^{2} \)
47 \( 1 + (-0.885 - 0.464i)T^{2} \)
53 \( 1 + (0.0825 - 0.183i)T + (-0.663 - 0.748i)T^{2} \)
59 \( 1 + (-0.0804 - 0.996i)T^{2} \)
61 \( 1 + (0.625 - 0.101i)T + (0.948 - 0.316i)T^{2} \)
67 \( 1 + (-0.0402 - 0.999i)T^{2} \)
71 \( 1 + (0.391 - 0.919i)T^{2} \)
73 \( 1 + (0.992 + 0.120i)T + (0.970 + 0.239i)T^{2} \)
79 \( 1 + (0.885 + 0.464i)T^{2} \)
83 \( 1 + (-0.120 + 0.992i)T^{2} \)
89 \( 1 + (0.153 + 0.574i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.879 - 1.39i)T + (-0.428 - 0.903i)T^{2} \)
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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

−8.868169403012954017764915421969, −8.178571233799724487824824807468, −7.00241224504764604333234688439, −6.22179937708757038930966773219, −5.50083874676020069521261157977, −4.70623254463894485142091610525, −4.08456457678021628154691582202, −2.92074139104356132760175531413, −2.02393964760878294101963492644, −1.33745021281992388345663138626, 1.19136304297289529340728242740, 2.69981684942423429934141164210, 3.54081366653472663043624851368, 4.42423853255553368683014956893, 5.32994189666849084114760076021, 6.03667962776935169117717133120, 6.50400212029819218707625049588, 7.30593116442081556118550370124, 8.048401975223792922003401171062, 8.969906221899529160065715648739

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