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Properties

Label 2-1170-65.9-c1-0-13
Degree $2$
Conductor $1170$
Sign $0.269 - 0.963i$
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 + (−1.23 − 1.86i)5-s + (−0.633 + 0.366i)7-s + 0.999i·8-s + (−0.133 − 2.23i)10-s + (−1.36 + 2.36i)11-s + (1.59 + 3.23i)13-s − 0.732·14-s + (−0.5 + 0.866i)16-s + (2.13 − 1.23i)17-s + (2.36 + 4.09i)19-s + (1 − 1.99i)20-s + (−2.36 + 1.36i)22-s + (3.63 + 2.09i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.550 − 0.834i)5-s + (−0.239 + 0.138i)7-s + 0.353i·8-s + (−0.0423 − 0.705i)10-s + (−0.411 + 0.713i)11-s + (0.443 + 0.896i)13-s − 0.195·14-s + (−0.125 + 0.216i)16-s + (0.517 − 0.298i)17-s + (0.542 + 0.940i)19-s + (0.223 − 0.447i)20-s + (−0.504 + 0.291i)22-s + (0.757 + 0.437i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.269 - 0.963i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.269 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.936657538\)
\(L(\frac12)\) \(\approx\) \(1.936657538\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 1.86i)T \)
13 \( 1 + (-1.59 - 3.23i)T \)
good7 \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.13 + 1.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.63 - 2.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.232 - 0.401i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-5.13 - 2.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.598 + 1.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.83 + 3.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.66iT - 47T^{2} \)
53 \( 1 - 4.26iT - 53T^{2} \)
59 \( 1 + (4.19 + 7.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.06 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.36 - 4.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.36 - 4.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 8.73iT - 83T^{2} \)
89 \( 1 + (4.46 - 7.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.693626817659373347394468870649, −9.173235508643727704971925443999, −8.045010923192711823487611626615, −7.57975704495053910827470819594, −6.55433294061970444224734652521, −5.60505334549253349932744948307, −4.76933861464750020343603521265, −4.02708952035897586744290618045, −2.95965644713662086357153955406, −1.41885545940776403824244272280, 0.74885441628253388092583773588, 2.70511100032771494173885148260, 3.22012798430521126699909847799, 4.21519596449018825694070747061, 5.36613768867231172047744065493, 6.15802647966606526006263248222, 7.05368692493333885705057971316, 7.86034553365508498819206546762, 8.740678143147222597489872897136, 9.940223585979703559798061340997

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