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Properties

Label 2-1170-13.10-c1-0-21
Degree $2$
Conductor $1170$
Sign $-0.835 - 0.549i$
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 − i·5-s + (−2.01 − 1.16i)7-s − 0.999i·8-s + (−0.5 − 0.866i)10-s + (−4.62 + 2.67i)11-s + (−3.60 − 0.161i)13-s − 2.32·14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (−3.48 − 2.01i)19-s + (−0.866 − 0.499i)20-s + (−2.67 + 4.62i)22-s + (2.46 + 4.27i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.760 − 0.438i)7-s − 0.353i·8-s + (−0.158 − 0.273i)10-s + (−1.39 + 0.805i)11-s + (−0.998 − 0.0447i)13-s − 0.620·14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (−0.799 − 0.461i)19-s + (−0.193 − 0.111i)20-s + (−0.569 + 0.986i)22-s + (0.514 + 0.891i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2043416398\)
\(L(\frac12)\) \(\approx\) \(0.2043416398\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (3.60 + 0.161i)T \)
good7 \( 1 + (2.01 + 1.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.62 - 2.67i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.48 + 2.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.46 - 4.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 3.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
37 \( 1 + (2.72 - 1.57i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.29 - 1.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.12 + 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.81iT - 47T^{2} \)
53 \( 1 - 5.48T + 53T^{2} \)
59 \( 1 + (5.87 + 3.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.267 - 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.55 + 2.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (13.7 + 7.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 7.96T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + (-1.50 + 0.869i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 + 8.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.509095316545186285498920514245, −8.535104739587982231069836620348, −7.42480959866007219865675739686, −6.84994450002385320682126801413, −5.66984033075497573361789618299, −4.90435130035079966410136593916, −4.10458087452454454105547224869, −2.92920981717755071749168568741, −1.95529948972394899007408723875, −0.06381862174671430237207036400, 2.61545850832663051529731483057, 2.87749098067409454654123833248, 4.32741783942862245354154128649, 5.23525649573034531488001184189, 6.07222633574874325964571625001, 6.81596786350400072184873815662, 7.67380064028201268806678930166, 8.493185381134590293896739492222, 9.433474920706726576734805654900, 10.38893460234287352382201372811

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