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Properties

Label 2-1170-65.49-c1-0-12
Degree $2$
Conductor $1170$
Sign $-0.575 - 0.818i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.10 + 0.767i)5-s + (−0.823 + 1.42i)7-s − 0.999·8-s + (0.385 + 2.20i)10-s + (−2.08 + 1.20i)11-s + (3.59 + 0.256i)13-s − 1.64·14-s + (−0.5 − 0.866i)16-s + (0.210 + 0.121i)17-s + (3.82 + 2.20i)19-s + (−1.71 + 1.43i)20-s + (−2.08 − 1.20i)22-s + (−7.46 + 4.31i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.939 + 0.343i)5-s + (−0.311 + 0.538i)7-s − 0.353·8-s + (0.121 + 0.696i)10-s + (−0.628 + 0.362i)11-s + (0.997 + 0.0710i)13-s − 0.439·14-s + (−0.125 − 0.216i)16-s + (0.0511 + 0.0295i)17-s + (0.877 + 0.506i)19-s + (−0.383 + 0.320i)20-s + (−0.444 − 0.256i)22-s + (−1.55 + 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.575 - 0.818i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.575 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.994378910\)
\(L(\frac12)\) \(\approx\) \(1.994378910\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-2.10 - 0.767i)T \)
13 \( 1 + (-3.59 - 0.256i)T \)
good7 \( 1 + (0.823 - 1.42i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.08 - 1.20i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.210 - 0.121i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.82 - 2.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.46 - 4.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0221 + 0.0383i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + (4.47 + 7.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.210 - 0.121i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.82 - 3.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.29T + 47T^{2} \)
53 \( 1 + 2.44iT - 53T^{2} \)
59 \( 1 + (-8.35 - 4.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.31 + 2.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.937 + 1.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.53 - 3.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.70T + 73T^{2} \)
79 \( 1 - 6.79T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + (-8.69 + 5.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.25 - 14.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.901688464654286698554356438618, −9.282612554424319914070342455625, −8.350535004993285125662513866323, −7.50029533224884958179444233762, −6.58345399555643376052988613024, −5.73399486383997473777518814091, −5.38495816592717181401152897647, −3.95320979751686064100503441873, −2.97175433987006427778800953681, −1.77283129417740455307121007378, 0.77412848824680711667642982152, 2.07499518644871142967526398762, 3.16685733333473537368271531599, 4.18092738695458390616856052013, 5.22731334135418028854095719804, 5.96449467566893958967691105106, 6.75605259212561570694525922413, 8.057053231786002265053063163074, 8.788029354960732904597030565009, 9.800769808268948755572542244901

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