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Properties

Label 2-3380-3380.307-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.985 - 0.171i$
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  + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.885 − 0.464i)5-s + (−0.464 + 0.885i)8-s + (−0.935 + 0.354i)9-s + (−0.663 + 0.748i)10-s + (−0.885 + 0.464i)13-s + (0.120 − 0.992i)16-s + (0.115 − 0.0359i)17-s + (0.748 − 0.663i)18-s + (0.354 − 0.935i)20-s + (0.568 − 0.822i)25-s + (0.663 − 0.748i)26-s + (1.06 − 0.402i)29-s + (0.239 + 0.970i)32-s + ⋯
L(s)  = 1  + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.885 − 0.464i)5-s + (−0.464 + 0.885i)8-s + (−0.935 + 0.354i)9-s + (−0.663 + 0.748i)10-s + (−0.885 + 0.464i)13-s + (0.120 − 0.992i)16-s + (0.115 − 0.0359i)17-s + (0.748 − 0.663i)18-s + (0.354 − 0.935i)20-s + (0.568 − 0.822i)25-s + (0.663 − 0.748i)26-s + (1.06 − 0.402i)29-s + (0.239 + 0.970i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.985 - 0.171i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8908664163\)
\(L(\frac12)\) \(\approx\) \(0.8908664163\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.935 - 0.354i)T \)
5 \( 1 + (-0.885 + 0.464i)T \)
13 \( 1 + (0.885 - 0.464i)T \)
good3 \( 1 + (0.935 - 0.354i)T^{2} \)
7 \( 1 + (-0.568 - 0.822i)T^{2} \)
11 \( 1 + (0.663 - 0.748i)T^{2} \)
17 \( 1 + (-0.115 + 0.0359i)T + (0.822 - 0.568i)T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-1.06 + 0.402i)T + (0.748 - 0.663i)T^{2} \)
31 \( 1 + (0.464 - 0.885i)T^{2} \)
37 \( 1 + (-1.92 - 0.475i)T + (0.885 + 0.464i)T^{2} \)
41 \( 1 + (-1.79 + 0.328i)T + (0.935 - 0.354i)T^{2} \)
43 \( 1 + (-0.464 - 0.885i)T^{2} \)
47 \( 1 + (-0.120 - 0.992i)T^{2} \)
53 \( 1 + (0.568 - 0.177i)T + (0.822 - 0.568i)T^{2} \)
59 \( 1 + (0.239 + 0.970i)T^{2} \)
61 \( 1 + (-1.45 + 0.764i)T + (0.568 - 0.822i)T^{2} \)
67 \( 1 + (0.120 + 0.992i)T^{2} \)
71 \( 1 + (-0.935 + 0.354i)T^{2} \)
73 \( 1 + (-1.87 - 0.709i)T + (0.748 + 0.663i)T^{2} \)
79 \( 1 + (0.120 + 0.992i)T^{2} \)
83 \( 1 + (0.354 - 0.935i)T^{2} \)
89 \( 1 + (1.11 + 1.11i)T + iT^{2} \)
97 \( 1 + (0.475 + 0.0576i)T + (0.970 + 0.239i)T^{2} \)
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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

−8.810146206738040642984479043124, −8.168989155619186752323619281645, −7.50517149313933007848310126550, −6.52000993580943529452158137974, −5.96143197815824403722848521834, −5.24243393074201876880933845682, −4.45543190470708623142114101664, −2.71248043394388756040669940782, −2.28181715272679535165130325282, −0.954421169179231213078517927015, 0.958183381625281257571944587662, 2.46154483093829531895306560233, 2.69982647011002515166871194723, 3.81676454384793450150334313162, 5.13891744384200975122730772757, 5.99610411590249823069117552818, 6.56810319259106597379990445878, 7.43403664280448511530001433002, 8.112861240413692949132994215211, 8.916270551310198475820805471576

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