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Properties

Label 2-1170-13.10-c1-0-8
Degree $2$
Conductor $1170$
Sign $0.711 + 0.702i$
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 + (−1.5 − 0.866i)7-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (−2.59 + 1.5i)11-s + (3.5 + 0.866i)13-s + 1.73·14-s + (−0.5 − 0.866i)16-s + (4.5 + 2.59i)19-s + (−0.866 − 0.499i)20-s + (1.5 − 2.59i)22-s + (−1.73 − 3i)23-s − 25-s + (−3.46 + i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.566 − 0.327i)7-s + 0.353i·8-s + (0.158 + 0.273i)10-s + (−0.783 + 0.452i)11-s + (0.970 + 0.240i)13-s + 0.462·14-s + (−0.125 − 0.216i)16-s + (1.03 + 0.596i)19-s + (−0.193 − 0.111i)20-s + (0.319 − 0.553i)22-s + (−0.361 − 0.625i)23-s − 0.200·25-s + (−0.679 + 0.196i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.711 + 0.702i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9984604276\)
\(L(\frac12)\) \(\approx\) \(0.9984604276\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good7 \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.46 - 6i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.19 + 3i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + (10.3 + 6i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-12.9 + 7.5i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.571298852525287762854224248979, −8.932452194567900184339047242677, −7.950459722789299333639445937846, −7.42093536262592766782674520389, −6.33251606732403172073131024762, −5.65860325515406083485264027347, −4.55815013585362866585935570210, −3.46179666406888936734040917164, −2.07041439750048115798518459712, −0.62533679537498635433639099711, 1.13799281355123868310687181497, 2.81706533892856470132188510100, 3.21100186546361693844877729837, 4.65088387459145165618235996187, 5.93591200514890429025625800234, 6.48040843868636936986873011333, 7.74455657077361876497136729216, 8.121656299923090509872913887686, 9.355661193052359836038954776649, 9.670716998525062474296023510104

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