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Properties

Label 2-3380-3380.3187-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.0268 + 0.999i$
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  + (0.200 − 0.979i)2-s + (−0.919 − 0.391i)4-s + (−0.534 + 0.845i)5-s + (−0.568 + 0.822i)8-s + (−0.316 − 0.948i)9-s + (0.721 + 0.692i)10-s + (0.903 + 0.428i)13-s + (0.692 + 0.721i)16-s + (−0.703 − 0.250i)17-s + (−0.992 + 0.120i)18-s + (0.822 − 0.568i)20-s + (−0.428 − 0.903i)25-s + (0.600 − 0.799i)26-s + (1.23 + 0.253i)29-s + (0.845 − 0.534i)32-s + ⋯
L(s)  = 1  + (0.200 − 0.979i)2-s + (−0.919 − 0.391i)4-s + (−0.534 + 0.845i)5-s + (−0.568 + 0.822i)8-s + (−0.316 − 0.948i)9-s + (0.721 + 0.692i)10-s + (0.903 + 0.428i)13-s + (0.692 + 0.721i)16-s + (−0.703 − 0.250i)17-s + (−0.992 + 0.120i)18-s + (0.822 − 0.568i)20-s + (−0.428 − 0.903i)25-s + (0.600 − 0.799i)26-s + (1.23 + 0.253i)29-s + (0.845 − 0.534i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0268 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0268 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.0268 + 0.999i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.0268 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066371319\)
\(L(\frac12)\) \(\approx\) \(1.066371319\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.200 + 0.979i)T \)
5 \( 1 + (0.534 - 0.845i)T \)
13 \( 1 + (-0.903 - 0.428i)T \)
good3 \( 1 + (0.316 + 0.948i)T^{2} \)
7 \( 1 + (-0.632 + 0.774i)T^{2} \)
11 \( 1 + (0.600 - 0.799i)T^{2} \)
17 \( 1 + (0.703 + 0.250i)T + (0.774 + 0.632i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-1.23 - 0.253i)T + (0.919 + 0.391i)T^{2} \)
31 \( 1 + (0.822 + 0.568i)T^{2} \)
37 \( 1 + (-1.02 + 1.62i)T + (-0.428 - 0.903i)T^{2} \)
41 \( 1 + (-0.975 + 1.35i)T + (-0.316 - 0.948i)T^{2} \)
43 \( 1 + (-0.903 - 0.428i)T^{2} \)
47 \( 1 + (-0.970 - 0.239i)T^{2} \)
53 \( 1 + (-0.339 - 1.85i)T + (-0.935 + 0.354i)T^{2} \)
59 \( 1 + (0.999 - 0.0402i)T^{2} \)
61 \( 1 + (-0.319 + 0.0258i)T + (0.987 - 0.160i)T^{2} \)
67 \( 1 + (-0.692 + 0.721i)T^{2} \)
71 \( 1 + (-0.979 - 0.200i)T^{2} \)
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 \( 1 + (-0.999 - 0.267i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.258 + 0.892i)T + (-0.845 - 0.534i)T^{2} \)
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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

−8.932012816051073091627282559130, −8.023563661808170313792943998249, −7.03614379320019572758827457789, −6.28598479541435626255189286848, −5.61840127146233591716119984819, −4.30208844915894000272855553069, −3.90343699460373826805884254993, −3.01496112518061626407283979631, −2.24685902303915678973957487184, −0.76428041502108337758883291710, 1.06376556184009747115585932596, 2.71473327118721693943155475413, 3.81228830768120093468912864602, 4.59008235026103539574024519469, 5.11093100975408454835048875437, 6.01238812234143537500538772491, 6.64040658127080628914306838063, 7.73681860988778537966509064309, 8.186168471159595661019465622664, 8.583804982016351843117346062468

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