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Properties

Label 2-315-9.4-c1-0-23
Degree $2$
Conductor $315$
Sign $-0.990 + 0.138i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (−0.631 − 1.09i)2-s + (0.478 − 1.66i)3-s + (0.202 − 0.350i)4-s + (0.5 − 0.866i)5-s + (−2.12 + 0.528i)6-s + (−0.5 − 0.866i)7-s − 3.03·8-s + (−2.54 − 1.59i)9-s − 1.26·10-s + (1.18 + 2.06i)11-s + (−0.487 − 0.505i)12-s + (0.892 − 1.54i)13-s + (−0.631 + 1.09i)14-s + (−1.20 − 1.24i)15-s + (1.51 + 2.62i)16-s − 2.57·17-s + ⋯
L(s)  = 1  + (−0.446 − 0.773i)2-s + (0.276 − 0.961i)3-s + (0.101 − 0.175i)4-s + (0.223 − 0.387i)5-s + (−0.866 + 0.215i)6-s + (−0.188 − 0.327i)7-s − 1.07·8-s + (−0.847 − 0.530i)9-s − 0.399·10-s + (0.358 + 0.621i)11-s + (−0.140 − 0.145i)12-s + (0.247 − 0.428i)13-s + (−0.168 + 0.292i)14-s + (−0.310 − 0.321i)15-s + (0.378 + 0.655i)16-s − 0.624·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.990 + 0.138i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0766106 - 1.10285i\)
\(L(\frac12)\) \(\approx\) \(0.0766106 - 1.10285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.478 + 1.66i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.631 + 1.09i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.18 - 2.06i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.892 + 1.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
19 \( 1 - 7.50T + 19T^{2} \)
23 \( 1 + (-0.0306 + 0.0531i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.47 + 2.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.42 - 2.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.89T + 37T^{2} \)
41 \( 1 + (3.02 - 5.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.06 + 8.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.84 + 3.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.33T + 53T^{2} \)
59 \( 1 + (-5.80 + 10.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.51 - 9.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.07 + 7.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.54T + 71T^{2} \)
73 \( 1 - 4.26T + 73T^{2} \)
79 \( 1 + (2.92 + 5.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.865 + 1.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + (-0.956 - 1.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.44286686143008311389047704286, −10.18312499370189195783361966766, −9.438545175001774443654726006367, −8.570869771801550644188242690891, −7.38764891504858828990339292617, −6.44071904740917511110295784023, −5.34325448510855244173801077372, −3.44515209851641532748535404115, −2.13732916599175477839359064227, −0.924090984229914726615787925472, 2.76224709841038761072837710001, 3.75885353620101871192043855574, 5.38265067476566351051392792699, 6.29238821287459347751340395719, 7.38433267529330898417707890743, 8.465829523018012181403592866427, 9.190038376745372246942813068755, 9.893038028631988435789109790499, 11.25761011215428050328535233880, 11.69778739958231256589101998995

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