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Properties

Label 2-1170-13.10-c1-0-7
Degree $2$
Conductor $1170$
Sign $0.992 - 0.125i$
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 + (3.84 + 2.21i)7-s − 0.999i·8-s + (0.5 + 0.866i)10-s + (−2.59 + 1.5i)11-s + (2.84 − 2.21i)13-s + 4.43·14-s + (−0.5 − 0.866i)16-s + (0.842 + 0.486i)19-s + (0.866 + 0.499i)20-s + (−1.5 + 2.59i)22-s + (−0.379 − 0.657i)23-s − 25-s + (1.35 − 3.34i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (1.45 + 0.838i)7-s − 0.353i·8-s + (0.158 + 0.273i)10-s + (−0.783 + 0.452i)11-s + (0.788 − 0.615i)13-s + 1.18·14-s + (−0.125 − 0.216i)16-s + (0.193 + 0.111i)19-s + (0.193 + 0.111i)20-s + (−0.319 + 0.553i)22-s + (−0.0791 − 0.137i)23-s − 0.200·25-s + (0.265 − 0.655i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.992 - 0.125i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.710455925\)
\(L(\frac12)\) \(\approx\) \(2.710455925\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.84 + 2.21i)T \)
good7 \( 1 + (-3.84 - 2.21i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.842 - 0.486i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.379 + 0.657i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.81 - 8.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.19 + 3i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.34 + 2.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + (4.05 + 2.34i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.68 + 9.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 6.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 6.68T + 79T^{2} \)
83 \( 1 - 9.36iT - 83T^{2} \)
89 \( 1 + (-6.65 + 3.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.68 - 2.70i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.08323419218908790188464313284, −8.851755395728210768183958026037, −8.197002708419622183410862391760, −7.33332246658367662778534519738, −6.25966351454617080601039051427, −5.25135480907509773563637958334, −4.87495189581247929853717383082, −3.50730241291343444998008049104, −2.52542163879797990060104301617, −1.50105639588299219924528123559, 1.12606075147418218073247488769, 2.48193923042325610240441547074, 4.00268389201075247549068559713, 4.49991544999383341160470037562, 5.44331468318171864889409979287, 6.26284023375636353370758314811, 7.44317619864792240230244635387, 8.010472858482419238366906568742, 8.614931478312890726266604331379, 9.791338108907639479428768461589

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