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Properties

Label 2-3380-3380.1783-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.405 - 0.914i$
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  + (0.534 − 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.919 − 0.391i)5-s + (−0.992 − 0.120i)8-s + (−0.999 − 0.0402i)9-s + (−0.822 + 0.568i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (0.0943 + 0.664i)17-s + (−0.568 + 0.822i)18-s + (0.0402 + 0.999i)20-s + (0.692 + 0.721i)25-s + (0.0804 + 0.996i)26-s + (−0.297 + 0.470i)29-s + (0.316 + 0.948i)32-s + ⋯
L(s)  = 1  + (0.534 − 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.919 − 0.391i)5-s + (−0.992 − 0.120i)8-s + (−0.999 − 0.0402i)9-s + (−0.822 + 0.568i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (0.0943 + 0.664i)17-s + (−0.568 + 0.822i)18-s + (0.0402 + 0.999i)20-s + (0.692 + 0.721i)25-s + (0.0804 + 0.996i)26-s + (−0.297 + 0.470i)29-s + (0.316 + 0.948i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.405 - 0.914i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.405 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2757685288\)
\(L(\frac12)\) \(\approx\) \(0.2757685288\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.534 + 0.845i)T \)
5 \( 1 + (0.919 + 0.391i)T \)
13 \( 1 + (0.799 - 0.600i)T \)
good3 \( 1 + (0.999 + 0.0402i)T^{2} \)
7 \( 1 + (-0.278 - 0.960i)T^{2} \)
11 \( 1 + (0.0804 + 0.996i)T^{2} \)
17 \( 1 + (-0.0943 - 0.664i)T + (-0.960 + 0.278i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.297 - 0.470i)T + (-0.428 - 0.903i)T^{2} \)
31 \( 1 + (0.992 + 0.120i)T^{2} \)
37 \( 1 + (0.304 + 0.101i)T + (0.799 + 0.600i)T^{2} \)
41 \( 1 + (-0.439 - 0.00884i)T + (0.999 + 0.0402i)T^{2} \)
43 \( 1 + (0.600 + 0.799i)T^{2} \)
47 \( 1 + (0.354 - 0.935i)T^{2} \)
53 \( 1 + (1.55 - 1.22i)T + (0.239 - 0.970i)T^{2} \)
59 \( 1 + (-0.979 - 0.200i)T^{2} \)
61 \( 1 + (1.32 + 0.565i)T + (0.692 + 0.721i)T^{2} \)
67 \( 1 + (-0.632 - 0.774i)T^{2} \)
71 \( 1 + (0.534 - 0.845i)T^{2} \)
73 \( 1 + (-0.464 + 0.885i)T + (-0.568 - 0.822i)T^{2} \)
79 \( 1 + (-0.354 + 0.935i)T^{2} \)
83 \( 1 + (-0.885 - 0.464i)T^{2} \)
89 \( 1 + (1.76 + 0.472i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.212 - 1.30i)T + (-0.948 - 0.316i)T^{2} \)
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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

−9.114351585691209625562781983198, −8.299078555589038374759757508558, −7.55524780836600664312530674609, −6.52591888500822064057927379423, −5.71752768995417369536250245546, −4.90495805437735642310361038498, −4.26092824489324639468597860051, −3.40897122655715972357018525892, −2.63647607182236538811008897938, −1.44588185845495715952274669371, 0.13446769328246180923009382043, 2.59540721424033198790809179145, 3.17812570525755094307484765428, 4.06920342430244478917576531504, 4.96703567747096738116153353402, 5.56598451563630862858377103811, 6.48855944822224781540284585039, 7.18154108047006003339616111638, 7.86227291349928958228087574631, 8.317601498047720360939864515232

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