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Properties

Label 2-315-7.2-c1-0-4
Degree $2$
Conductor $315$
Sign $0.106 - 0.994i$
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  + (−0.933 + 1.61i)2-s + (−0.741 − 1.28i)4-s + (0.5 − 0.866i)5-s + (1.69 − 2.03i)7-s − 0.965·8-s + (0.933 + 1.61i)10-s + (2.67 + 4.63i)11-s + 4.34·13-s + (1.70 + 4.63i)14-s + (2.38 − 4.12i)16-s + (0.933 + 1.61i)17-s + (−2.62 + 4.54i)19-s − 1.48·20-s − 9.98·22-s + (3.45 − 5.97i)23-s + ⋯
L(s)  = 1  + (−0.659 + 1.14i)2-s + (−0.370 − 0.642i)4-s + (0.223 − 0.387i)5-s + (0.639 − 0.768i)7-s − 0.341·8-s + (0.295 + 0.511i)10-s + (0.806 + 1.39i)11-s + 1.20·13-s + (0.456 + 1.23i)14-s + (0.595 − 1.03i)16-s + (0.226 + 0.391i)17-s + (−0.602 + 1.04i)19-s − 0.331·20-s − 2.12·22-s + (0.719 − 1.24i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.106 - 0.994i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.106 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.799262 + 0.718175i\)
\(L(\frac12)\) \(\approx\) \(0.799262 + 0.718175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.69 + 2.03i)T \)
good2 \( 1 + (0.933 - 1.61i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-2.67 - 4.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
17 \( 1 + (-0.933 - 1.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.62 - 4.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.45 + 5.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + (-4.71 - 8.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.08 - 1.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.314T + 41T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 + (-4.21 + 7.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.86 + 3.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.639 - 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.39 + 2.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.04 + 1.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.81T + 71T^{2} \)
73 \( 1 + (-2.81 - 4.87i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.97 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + (1.45 - 2.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.91T + 97T^{2} \)
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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

−11.95568143600531203312068170971, −10.65310631756579900562538643947, −9.831867845392868306208696348034, −8.687244253046357259306986766380, −8.185598797300372359698041980818, −7.01004962990889447122702604235, −6.39690462270030916896670660877, −5.05317183826240722426661122438, −3.87226664594060768847993051839, −1.48079211008891590948165936086, 1.22560875261141135916943233869, 2.64717253933707739182651678021, 3.72941460144804020083056790605, 5.58687537756362826186630048867, 6.41382575404885248926124876058, 8.081799132798464839739551105371, 8.961563102147089636423446473291, 9.457440732820602270736055068187, 10.92644089659957362301987526819, 11.22238628809027721646965638196

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