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Properties

Label 2-1170-65.29-c1-0-32
Degree $2$
Conductor $1170$
Sign $-0.764 + 0.644i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.539 − 2.17i)5-s + (0.614 + 0.354i)7-s − 0.999i·8-s + (−0.618 − 2.14i)10-s + (−2.25 − 3.90i)11-s + (−3.35 − 1.30i)13-s + 0.709·14-s + (−0.5 − 0.866i)16-s + (−2.74 − 1.58i)17-s + (−0.0603 + 0.104i)19-s + (−1.60 − 1.55i)20-s + (−3.90 − 2.25i)22-s + (−4.30 + 2.48i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.241 − 0.970i)5-s + (0.232 + 0.134i)7-s − 0.353i·8-s + (−0.195 − 0.679i)10-s + (−0.679 − 1.17i)11-s + (−0.931 − 0.362i)13-s + 0.189·14-s + (−0.125 − 0.216i)16-s + (−0.665 − 0.384i)17-s + (−0.0138 + 0.0239i)19-s + (−0.359 − 0.347i)20-s + (−0.832 − 0.480i)22-s + (−0.897 + 0.518i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.873208414\)
\(L(\frac12)\) \(\approx\) \(1.873208414\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-0.539 + 2.17i)T \)
13 \( 1 + (3.35 + 1.30i)T \)
good7 \( 1 + (-0.614 - 0.354i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.74 + 1.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0603 - 0.104i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.30 - 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.63 - 6.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.66T + 31T^{2} \)
37 \( 1 + (-6.02 + 3.47i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.223 - 0.387i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.73 + i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.70iT - 47T^{2} \)
53 \( 1 + 9.58iT - 53T^{2} \)
59 \( 1 + (-2.87 + 4.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.53 - 6.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.53 + 1.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.09 - 7.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.74iT - 73T^{2} \)
79 \( 1 - 16.0T + 79T^{2} \)
83 \( 1 - 0.355iT - 83T^{2} \)
89 \( 1 + (1.81 + 3.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.84 + 3.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.630513608891252992089805124029, −8.538155710825457364244140350363, −8.051334328387407018102254679997, −6.80468586270953271130045424013, −5.74875834605989715297870264915, −5.15198126247419089605080876046, −4.38448356493505889017441137322, −3.12252371769163381144893109110, −2.11501578969457314615871916018, −0.61950168314985686670500324058, 2.16349248698896427236167823311, 2.78265838366730912576291209285, 4.31791173441951472181148459969, 4.73969891587132951234438348916, 6.12847850284352550756474964617, 6.59934655904060940361063650280, 7.61050848993973962290559057178, 8.034747915882018028922277550447, 9.490122967161491386411764922753, 10.13197629052358333545414496478

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