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Properties

Label 2-1170-39.2-c1-0-5
Degree $2$
Conductor $1170$
Sign $-0.677 - 0.735i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 − 0.707i)5-s + (−1.03 + 0.277i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.105 − 0.0282i)11-s + (1.19 + 3.40i)13-s − 1.07i·14-s + (0.500 + 0.866i)16-s + (1.09 − 1.90i)17-s + (0.578 + 2.15i)19-s + (0.258 + 0.965i)20-s + (0.0545 − 0.0944i)22-s + (−1.34 − 2.32i)23-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.316 − 0.316i)5-s + (−0.391 + 0.104i)7-s + (0.249 − 0.249i)8-s + (0.273 − 0.158i)10-s + (−0.0317 − 0.00850i)11-s + (0.332 + 0.943i)13-s − 0.286i·14-s + (0.125 + 0.216i)16-s + (0.266 − 0.461i)17-s + (0.132 + 0.495i)19-s + (0.0578 + 0.215i)20-s + (0.0116 − 0.0201i)22-s + (−0.279 − 0.483i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.677 - 0.735i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8755025194\)
\(L(\frac12)\) \(\approx\) \(0.8755025194\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-1.19 - 3.40i)T \)
good7 \( 1 + (1.03 - 0.277i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.105 + 0.0282i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.09 + 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.578 - 2.15i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.34 + 2.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.745 - 0.430i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.123 - 0.123i)T - 31iT^{2} \)
37 \( 1 + (2.82 - 10.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.03 - 3.86i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.5 - 6.07i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.91 - 8.91i)T - 47iT^{2} \)
53 \( 1 - 3.57iT - 53T^{2} \)
59 \( 1 + (2.10 + 7.85i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.30 - 7.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.67 + 0.985i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.12 - 0.838i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-9.26 - 9.26i)T + 73iT^{2} \)
79 \( 1 - 3.58T + 79T^{2} \)
83 \( 1 + (-6.23 - 6.23i)T + 83iT^{2} \)
89 \( 1 + (0.486 + 0.130i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.35 + 5.06i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.731127812450906205094949045497, −9.289340794521339213531228736430, −8.316911516540805661934173270463, −7.71890379386115479497805234166, −6.66852193504417991223130445557, −6.10314210889104071253681112798, −4.96326780747077436692698736351, −4.19573833929951627928699087783, −3.01749864074123397655148492236, −1.34865119455159528042619857791, 0.43280212181729393034915715469, 2.04358910751604933105268561199, 3.27038215782715482645027458410, 3.85467847891115700805749621906, 5.14828462863894112875078193332, 6.04473484377267223932124424924, 7.18154817668019740071124122040, 7.891091423600318388637280502348, 8.781725177489377436765067323819, 9.571101017447918117586370443753

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