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Properties

Label 2-3380-3380.1943-c0-0-0
Degree $2$
Conductor $3380$
Sign $-0.449 - 0.893i$
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  + (0.721 + 0.692i)2-s + (0.0402 + 0.999i)4-s + (0.948 + 0.316i)5-s + (−0.663 + 0.748i)8-s + (−0.960 + 0.278i)9-s + (0.464 + 0.885i)10-s + (0.200 + 0.979i)13-s + (−0.996 + 0.0804i)16-s + (1.13 − 0.747i)17-s + (−0.885 − 0.464i)18-s + (−0.278 + 0.960i)20-s + (0.799 + 0.600i)25-s + (−0.534 + 0.845i)26-s + (1.32 + 1.27i)29-s + (−0.774 − 0.632i)32-s + ⋯
L(s)  = 1  + (0.721 + 0.692i)2-s + (0.0402 + 0.999i)4-s + (0.948 + 0.316i)5-s + (−0.663 + 0.748i)8-s + (−0.960 + 0.278i)9-s + (0.464 + 0.885i)10-s + (0.200 + 0.979i)13-s + (−0.996 + 0.0804i)16-s + (1.13 − 0.747i)17-s + (−0.885 − 0.464i)18-s + (−0.278 + 0.960i)20-s + (0.799 + 0.600i)25-s + (−0.534 + 0.845i)26-s + (1.32 + 1.27i)29-s + (−0.774 − 0.632i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.449 - 0.893i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.449 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.021906053\)
\(L(\frac12)\) \(\approx\) \(2.021906053\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.721 - 0.692i)T \)
5 \( 1 + (-0.948 - 0.316i)T \)
13 \( 1 + (-0.200 - 0.979i)T \)
good3 \( 1 + (0.960 - 0.278i)T^{2} \)
7 \( 1 + (0.919 + 0.391i)T^{2} \)
11 \( 1 + (-0.534 + 0.845i)T^{2} \)
17 \( 1 + (-1.13 + 0.747i)T + (0.391 - 0.919i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-1.32 - 1.27i)T + (0.0402 + 0.999i)T^{2} \)
31 \( 1 + (0.663 - 0.748i)T^{2} \)
37 \( 1 + (1.14 + 1.39i)T + (-0.200 + 0.979i)T^{2} \)
41 \( 1 + (1.97 - 0.280i)T + (0.960 - 0.278i)T^{2} \)
43 \( 1 + (0.979 - 0.200i)T^{2} \)
47 \( 1 + (-0.568 + 0.822i)T^{2} \)
53 \( 1 + (-0.0665 - 1.10i)T + (-0.992 + 0.120i)T^{2} \)
59 \( 1 + (-0.160 + 0.987i)T^{2} \)
61 \( 1 + (-1.13 - 0.380i)T + (0.799 + 0.600i)T^{2} \)
67 \( 1 + (-0.996 - 0.0804i)T^{2} \)
71 \( 1 + (0.721 + 0.692i)T^{2} \)
73 \( 1 + (-0.239 - 0.970i)T + (-0.885 + 0.464i)T^{2} \)
79 \( 1 + (0.568 - 0.822i)T^{2} \)
83 \( 1 + (0.970 - 0.239i)T^{2} \)
89 \( 1 + (0.509 + 1.90i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.68 - 0.801i)T + (0.632 + 0.774i)T^{2} \)
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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

−8.783825461286258577620672528187, −8.372729685312674665573964694778, −7.19079345585946463694836181793, −6.81716835904943913586617159552, −5.92052251173104830206267617641, −5.34960937184611596950671425750, −4.73106617975573261629090298565, −3.44579299226999555233870314186, −2.86976580715254371102610998504, −1.80359654744062276631766253727, 0.966580319423267674531496129737, 2.05035097883085074682985395666, 3.04266590033708584959506193997, 3.60021963669278028859034954708, 4.96967619804830892924663684446, 5.32347726324335895129651481780, 6.18934123587932833299665823803, 6.56486393659234904679652882085, 8.107084662307093720540311810950, 8.543195663713193394067720587612

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