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Properties

Label 2-315-9.4-c1-0-6
Degree $2$
Conductor $315$
Sign $-0.710 - 0.703i$
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  + (1.38 + 2.40i)2-s + (1.16 − 1.27i)3-s + (−2.85 + 4.95i)4-s + (−0.5 + 0.866i)5-s + (4.69 + 1.03i)6-s + (0.5 + 0.866i)7-s − 10.3·8-s + (−0.275 − 2.98i)9-s − 2.77·10-s + (0.923 + 1.59i)11-s + (2.99 + 9.43i)12-s + (−1.16 + 2.02i)13-s + (−1.38 + 2.40i)14-s + (0.524 + 1.65i)15-s + (−8.62 − 14.9i)16-s + 5.53·17-s + ⋯
L(s)  = 1  + (0.982 + 1.70i)2-s + (0.673 − 0.738i)3-s + (−1.42 + 2.47i)4-s + (−0.223 + 0.387i)5-s + (1.91 + 0.420i)6-s + (0.188 + 0.327i)7-s − 3.65·8-s + (−0.0916 − 0.995i)9-s − 0.878·10-s + (0.278 + 0.482i)11-s + (0.865 + 2.72i)12-s + (−0.323 + 0.560i)13-s + (−0.371 + 0.642i)14-s + (0.135 + 0.426i)15-s + (−2.15 − 3.73i)16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.710 - 0.703i$
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: \((2,\ 315,\ (\ :1/2),\ -0.710 - 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.858614 + 2.08627i\)
\(L(\frac12)\) \(\approx\) \(0.858614 + 2.08627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.16 + 1.27i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1.38 - 2.40i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-0.923 - 1.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.16 - 2.02i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 - 4.34T + 19T^{2} \)
23 \( 1 + (-3.91 + 6.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.39 + 5.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.962 - 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.33T + 37T^{2} \)
41 \( 1 + (-0.673 + 1.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.20 - 2.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.45 + 4.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.98T + 53T^{2} \)
59 \( 1 + (0.606 - 1.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.95 - 3.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.57 - 9.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.81T + 71T^{2} \)
73 \( 1 - 4.63T + 73T^{2} \)
79 \( 1 + (-0.313 - 0.542i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.70 - 2.96i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + (-2.49 - 4.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.27865595696337231731845781660, −11.81547945966188520943885689847, −9.609114488791030268945354999510, −8.703239647861365431240148000539, −7.77240437057530963609689392363, −7.17302636345094569549424583082, −6.35547138008096469259390149908, −5.23972745730188021656292359137, −3.96870987330984808528190590572, −2.85306029586688379401149233907, 1.39241502448521232981972147789, 3.16963800554142448498586520209, 3.62155823014071574596114485223, 4.97646566339322848397670259700, 5.50919236445894114023094533332, 7.73651926839959652861082536024, 9.100191722255301508582858259846, 9.653607094759627781835297520989, 10.58090460612077415104874297692, 11.29920913210195571711569327538

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