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Properties

Label 2-315-35.27-c1-0-15
Degree $2$
Conductor $315$
Sign $-0.945 + 0.326i$
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  + (−0.707 − 0.707i)2-s − 0.999i·4-s − 2.23i·5-s + (0.581 − 2.58i)7-s + (−2.12 + 2.12i)8-s + (−1.58 + 1.58i)10-s − 4.24·11-s + (3.16 + 3.16i)13-s + (−2.23 + 1.41i)14-s + 1.00·16-s + (2.23 − 2.23i)17-s − 3.16·19-s − 2.23·20-s + (3 + 3i)22-s + (−1.41 + 1.41i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s − 0.499i·4-s − 0.999i·5-s + (0.219 − 0.975i)7-s + (−0.750 + 0.750i)8-s + (−0.500 + 0.500i)10-s − 1.27·11-s + (0.877 + 0.877i)13-s + (−0.597 + 0.377i)14-s + 0.250·16-s + (0.542 − 0.542i)17-s − 0.725·19-s − 0.499·20-s + (0.639 + 0.639i)22-s + (−0.294 + 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.945 + 0.326i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131732 - 0.785982i\)
\(L(\frac12)\) \(\approx\) \(0.131732 - 0.785982i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
7 \( 1 + (-0.581 + 2.58i)T \)
good2 \( 1 + (0.707 + 0.707i)T + 2iT^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + (-3.16 - 3.16i)T + 13iT^{2} \)
17 \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \)
19 \( 1 + 3.16T + 19T^{2} \)
23 \( 1 + (1.41 - 1.41i)T - 23iT^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 9.48iT - 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + (-8.94 + 8.94i)T - 47iT^{2} \)
53 \( 1 + (-7.07 + 7.07i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + (-3.16 - 3.16i)T + 73iT^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 + (-4.47 - 4.47i)T + 83iT^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (-3.16 + 3.16i)T - 97iT^{2} \)
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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

−11.20813794855650106841503537609, −10.20457479076663442472728518473, −9.583983586974076109214607974497, −8.488124627299701497187080466709, −7.73644166978417941877384545491, −6.19974234574008333346895095589, −5.14896710525280000379208156444, −4.06556031681943537257684683186, −2.10947101182841777143617389416, −0.67044928474312393554136277868, 2.57644769854430966341711731838, 3.54971607626692794962323264546, 5.44688629030237671134229664505, 6.33520969968046699570389825136, 7.46300654485046915001102654481, 8.240533127399427219047704232814, 8.943334029777357173508596279061, 10.38852886740383591015950435520, 10.83654782872076988401564298146, 12.25796942299049718153590244727

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