Properties

Label 2-1170-65.49-c1-0-30
Degree $2$
Conductor $1170$
Sign $0.886 + 0.462i$
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 + (1.40 − 1.74i)5-s + (0.763 − 1.32i)7-s − 0.999·8-s + (2.20 + 0.341i)10-s + (1.14 − 0.658i)11-s + (−2.41 − 2.67i)13-s + 1.52·14-s + (−0.5 − 0.866i)16-s + (1.35 + 0.784i)17-s + (−4.18 − 2.41i)19-s + (0.809 + 2.08i)20-s + (1.14 + 0.658i)22-s + (7.31 − 4.22i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.626 − 0.779i)5-s + (0.288 − 0.500i)7-s − 0.353·8-s + (0.698 + 0.107i)10-s + (0.343 − 0.198i)11-s + (−0.669 − 0.743i)13-s + 0.408·14-s + (−0.125 − 0.216i)16-s + (0.329 + 0.190i)17-s + (−0.959 − 0.553i)19-s + (0.180 + 0.466i)20-s + (0.243 + 0.140i)22-s + (1.52 − 0.880i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.886 + 0.462i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.051098740\)
\(L(\frac12)\) \(\approx\) \(2.051098740\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 1.74i)T \)
13 \( 1 + (2.41 + 2.67i)T \)
good7 \( 1 + (-0.763 + 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.14 + 0.658i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.35 - 0.784i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.18 + 2.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.31 + 4.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.21 - 3.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.62iT - 31T^{2} \)
37 \( 1 + (1.40 + 2.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.35 - 0.784i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.58 + 2.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.94T + 47T^{2} \)
53 \( 1 + 13.9iT - 53T^{2} \)
59 \( 1 + (-9.07 - 5.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.38 - 2.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.8 - 7.41i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.98T + 73T^{2} \)
79 \( 1 - 4.87T + 79T^{2} \)
83 \( 1 - 6.39T + 83T^{2} \)
89 \( 1 + (15.9 - 9.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.963 + 1.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.616116042385547291028847176103, −8.669845131920667781786987563208, −8.213297567725515577268728145206, −7.05454372451978550512378512103, −6.43899638140979424476103891569, −5.23782503008248984929688059493, −4.87586025355544235937477631136, −3.73836679784619936062073394633, −2.41530487476628785161269277169, −0.822804601017247955802596442316, 1.61939834840033834002722820316, 2.50151975036290588021930290276, 3.49772993743013033440577955993, 4.67913289380396669288130958794, 5.49590666577375432388728995147, 6.46538318145160059423121305571, 7.14644432169125632199245837374, 8.362747901052074956075775970623, 9.337762677839961221402848936543, 9.838022335850383829887268671576

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