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Properties

Label 2-3380-3380.3243-c0-0-0
Degree $2$
Conductor $3380$
Sign $-0.641 + 0.767i$
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  + (0.774 − 0.632i)2-s + (0.200 − 0.979i)4-s + (−0.0402 − 0.999i)5-s + (−0.464 − 0.885i)8-s + (0.160 + 0.987i)9-s + (−0.663 − 0.748i)10-s + (0.845 − 0.534i)13-s + (−0.919 − 0.391i)16-s + (−0.366 − 1.62i)17-s + (0.748 + 0.663i)18-s + (−0.987 − 0.160i)20-s + (−0.996 + 0.0804i)25-s + (0.316 − 0.948i)26-s + (−0.664 + 0.542i)29-s + (−0.960 + 0.278i)32-s + ⋯
L(s)  = 1  + (0.774 − 0.632i)2-s + (0.200 − 0.979i)4-s + (−0.0402 − 0.999i)5-s + (−0.464 − 0.885i)8-s + (0.160 + 0.987i)9-s + (−0.663 − 0.748i)10-s + (0.845 − 0.534i)13-s + (−0.919 − 0.391i)16-s + (−0.366 − 1.62i)17-s + (0.748 + 0.663i)18-s + (−0.987 − 0.160i)20-s + (−0.996 + 0.0804i)25-s + (0.316 − 0.948i)26-s + (−0.664 + 0.542i)29-s + (−0.960 + 0.278i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.641 + 0.767i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.641 + 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.867998870\)
\(L(\frac12)\) \(\approx\) \(1.867998870\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.774 + 0.632i)T \)
5 \( 1 + (0.0402 + 0.999i)T \)
13 \( 1 + (-0.845 + 0.534i)T \)
good3 \( 1 + (-0.160 - 0.987i)T^{2} \)
7 \( 1 + (-0.428 - 0.903i)T^{2} \)
11 \( 1 + (0.316 - 0.948i)T^{2} \)
17 \( 1 + (0.366 + 1.62i)T + (-0.903 + 0.428i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.664 - 0.542i)T + (0.200 - 0.979i)T^{2} \)
31 \( 1 + (0.464 + 0.885i)T^{2} \)
37 \( 1 + (-0.334 + 1.15i)T + (-0.845 - 0.534i)T^{2} \)
41 \( 1 + (-0.130 + 0.153i)T + (-0.160 - 0.987i)T^{2} \)
43 \( 1 + (-0.534 - 0.845i)T^{2} \)
47 \( 1 + (-0.120 + 0.992i)T^{2} \)
53 \( 1 + (-1.86 - 0.580i)T + (0.822 + 0.568i)T^{2} \)
59 \( 1 + (0.721 + 0.692i)T^{2} \)
61 \( 1 + (0.00648 + 0.160i)T + (-0.996 + 0.0804i)T^{2} \)
67 \( 1 + (-0.919 + 0.391i)T^{2} \)
71 \( 1 + (0.774 - 0.632i)T^{2} \)
73 \( 1 + (0.935 - 0.354i)T + (0.748 - 0.663i)T^{2} \)
79 \( 1 + (0.120 - 0.992i)T^{2} \)
83 \( 1 + (0.354 + 0.935i)T^{2} \)
89 \( 1 + (0.407 + 1.52i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.287 - 0.382i)T + (-0.278 + 0.960i)T^{2} \)
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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

−8.775363853738278624108199080201, −7.69124274387298610836767675940, −7.07220373789401115081513656396, −5.81990541386806623991885046571, −5.40694641991448651489396016996, −4.63473208934870836697681048363, −3.98962832959814312996269619202, −2.88976331190269556031589717552, −1.98342998747332175496855568421, −0.873607910284478883460163091851, 1.82967832119874803378546270520, 2.94849113645481769523470827445, 3.90732123636570199077118935886, 4.07309399285088047388409997005, 5.52559191859214527159594548514, 6.24443597166261156827559450352, 6.59770389706762548960287699551, 7.32688944835106023961223777752, 8.250735119391850851454459649202, 8.780978849147432984348870136855

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