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Properties

Label 2-1170-13.10-c1-0-5
Degree $2$
Conductor $1170$
Sign $0.252 - 0.967i$
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 + (1.73 + i)7-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (−0.401 + 0.232i)11-s + (1 + 3.46i)13-s − 1.99·14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (−0.464 − 0.267i)19-s + (−0.866 − 0.499i)20-s + (0.232 − 0.401i)22-s + (0.133 + 0.232i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (0.654 + 0.377i)7-s + 0.353i·8-s + (0.158 + 0.273i)10-s + (−0.121 + 0.0699i)11-s + (0.277 + 0.960i)13-s − 0.534·14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (−0.106 − 0.0614i)19-s + (−0.193 − 0.111i)20-s + (0.0494 − 0.0856i)22-s + (0.0279 + 0.0483i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.252 - 0.967i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.156704961\)
\(L(\frac12)\) \(\approx\) \(1.156704961\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 3.46i)T \)
good7 \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.401 - 0.232i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.464 + 0.267i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.133 - 0.232i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.86 - 3.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (-1.03 + 0.598i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.964 + 1.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + (-1.33 - 0.767i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.19 - 9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.92 + 2.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.26 - 4.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 0.0717T + 79T^{2} \)
83 \( 1 - 4.92iT - 83T^{2} \)
89 \( 1 + (6.46 - 3.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.46 + 3.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.797683162953039302980291705841, −8.838999315196501280295675122621, −8.566499464228931828325164403630, −7.59007756896831465309737004612, −6.69649808659619107169699267219, −5.83249634859988207577819728359, −4.90437790900253374430228014299, −3.99714080600462324151895101523, −2.34476944047085118245033778104, −1.31760104872918822249271101831, 0.67071398504075473525086385681, 2.15445974064712293644801157202, 3.15803039253591244165326822287, 4.26013888597885307488309349793, 5.34056642445280605196542758014, 6.43457967182876006210417657943, 7.35393202403011814644237997774, 8.003374919200840741263057586301, 8.751284948306920255960610846824, 9.757230484480373807378495215404

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