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Properties

Label 2-1170-65.9-c1-0-28
Degree $2$
Conductor $1170$
Sign $-0.273 + 0.961i$
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 + (−2.21 + 0.311i)5-s + (−3.38 + 1.95i)7-s + 0.999i·8-s + (−2.07 − 0.837i)10-s + (0.533 − 0.923i)11-s + (2.24 − 2.82i)13-s − 3.90·14-s + (−0.5 + 0.866i)16-s + (−1.13 + 0.655i)17-s + (−3.29 − 5.70i)19-s + (−1.37 − 1.76i)20-s + (0.923 − 0.533i)22-s + (−1.85 − 1.07i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.990 + 0.139i)5-s + (−1.27 + 0.737i)7-s + 0.353i·8-s + (−0.655 − 0.264i)10-s + (0.160 − 0.278i)11-s + (0.622 − 0.782i)13-s − 1.04·14-s + (−0.125 + 0.216i)16-s + (−0.275 + 0.158i)17-s + (−0.756 − 1.30i)19-s + (−0.307 − 0.394i)20-s + (0.196 − 0.113i)22-s + (−0.387 − 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.273 + 0.961i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4504845118\)
\(L(\frac12)\) \(\approx\) \(0.4504845118\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.311i)T \)
13 \( 1 + (-2.24 + 2.82i)T \)
good7 \( 1 + (3.38 - 1.95i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.533 + 0.923i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.13 - 0.655i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.29 + 5.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.85 + 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.52 + 7.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + (4.34 + 2.51i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.97 - 5.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.73 - i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.0967iT - 47T^{2} \)
53 \( 1 + 1.49iT - 53T^{2} \)
59 \( 1 + (3.59 + 6.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.94 + 6.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.29 - 4.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.73 + 13.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + 4.30T + 79T^{2} \)
83 \( 1 - 9.69iT - 83T^{2} \)
89 \( 1 + (2.26 - 3.91i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.68 - 2.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.370657054627710785829224374107, −8.571717357176626286660762249057, −7.916227914893700247486807808847, −6.73056781771327990341829351191, −6.35033314887792647011881835091, −5.32395470661664980592382064845, −4.19948343949209686481258277104, −3.37616389872499034977135214531, −2.57756502957774993921035105843, −0.15495540944133767918628527890, 1.56812181490474420298421531631, 3.24814121567961369825848065443, 3.81216792336027919954287820552, 4.52999584482149593058025592944, 5.82827092571880070651020104439, 6.77741453192537670451088078401, 7.23177594465335805965042869833, 8.480025796003507266348376394720, 9.231273386700996996425596812115, 10.31747071039290434140257984159

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