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Properties

Label 2-1170-13.10-c1-0-19
Degree $2$
Conductor $1170$
Sign $-0.823 + 0.566i$
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 + (−3.45 − 1.99i)7-s − 0.999i·8-s + (0.5 + 0.866i)10-s + (−0.142 + 0.0820i)11-s + (1.50 − 3.27i)13-s − 3.99·14-s + (−0.5 − 0.866i)16-s + (−0.783 + 1.35i)17-s + (−0.716 − 0.413i)19-s + (0.866 + 0.499i)20-s + (−0.0820 + 0.142i)22-s + (−3.49 − 6.05i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−1.30 − 0.754i)7-s − 0.353i·8-s + (0.158 + 0.273i)10-s + (−0.0428 + 0.0247i)11-s + (0.418 − 0.908i)13-s − 1.06·14-s + (−0.125 − 0.216i)16-s + (−0.190 + 0.329i)17-s + (−0.164 − 0.0948i)19-s + (0.193 + 0.111i)20-s + (−0.0175 + 0.0303i)22-s + (−0.728 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.823 + 0.566i$
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: \((2,\ 1170,\ (\ :1/2),\ -0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.225935568\)
\(L(\frac12)\) \(\approx\) \(1.225935568\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 + 3.27i)T \)
good7 \( 1 + (3.45 + 1.99i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.142 - 0.0820i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.783 - 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.716 + 0.413i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.49 + 6.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.92 + 5.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.0858iT - 31T^{2} \)
37 \( 1 + (7.24 - 4.18i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.48 - 4.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.62 + 9.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.31iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.21 + 5.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.2 + 7.65i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.179iT - 73T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + 1.23iT - 83T^{2} \)
89 \( 1 + (-8.98 + 5.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-16.3 - 9.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.862855434013022120625225647391, −8.663525184163299304167684238451, −7.67448151438853712235337410263, −6.60313330871626442116331890494, −6.27761429089652237044269608842, −5.11717955950380528047791772809, −3.87309584636142758465484626388, −3.37352754450002739137880632506, −2.22822112545568715150453948901, −0.40256962627669249103364398219, 1.89458322764184369146032785689, 3.18571803851456051404614758317, 3.95743118527469890293904314145, 5.13471612980753938887817141625, 5.90625832877612545404172265642, 6.61372174974082089637654912488, 7.45113345723985402985174388358, 8.589636926326621299382690570501, 9.221934306642340389734858121693, 9.871114613896143631153486691994

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