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Properties

Label 2-315-35.13-c1-0-9
Degree $2$
Conductor $315$
Sign $0.714 - 0.699i$
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  + (−1.58 + 1.58i)2-s − 3.00i·4-s + (2 + i)5-s + (−0.581 − 2.58i)7-s + (1.58 + 1.58i)8-s + (−4.74 + 1.58i)10-s + 3.16·11-s + (3.16 − 3.16i)13-s + (5 + 3.16i)14-s + 0.999·16-s + (−5 − 5i)17-s + 3.16·19-s + (3.00 − 6.00i)20-s + (−5.00 + 5.00i)22-s + (3.16 + 3.16i)23-s + ⋯
L(s)  = 1  + (−1.11 + 1.11i)2-s − 1.50i·4-s + (0.894 + 0.447i)5-s + (−0.219 − 0.975i)7-s + (0.559 + 0.559i)8-s + (−1.50 + 0.500i)10-s + 0.953·11-s + (0.877 − 0.877i)13-s + (1.33 + 0.845i)14-s + 0.249·16-s + (−1.21 − 1.21i)17-s + 0.725·19-s + (0.670 − 1.34i)20-s + (−1.06 + 1.06i)22-s + (0.659 + 0.659i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812542 + 0.331641i\)
\(L(\frac12)\) \(\approx\) \(0.812542 + 0.331641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
7 \( 1 + (0.581 + 2.58i)T \)
good2 \( 1 + (1.58 - 1.58i)T - 2iT^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \)
17 \( 1 + (5 + 5i)T + 17iT^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + (-3.16 - 3.16i)T + 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 3.16iT - 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (3.16 + 3.16i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 + (-8 + 8i)T - 67iT^{2} \)
71 \( 1 - 9.48T + 71T^{2} \)
73 \( 1 + (9.48 - 9.48i)T - 73iT^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + (-10 + 10i)T - 83iT^{2} \)
89 \( 1 + 10T + 89T^{2} \)
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

−11.31597067523380822762359302334, −10.54876832551089247730687256186, −9.558350741005886000179570052451, −9.104807410727220658884572336421, −7.85158196118557223083081791335, −6.87563143205460651470953359837, −6.42223239245630647515321923064, −5.17984715228071354813589173006, −3.33902897209618634307041500072, −1.11292529597884335352530308173, 1.45281163328353898407150477273, 2.43866985118941708211655608549, 4.02050814093234876396450240982, 5.72719865359402923409590223289, 6.66262625980810482660229570992, 8.493047775512827662398329074399, 8.966379870948371071062621234578, 9.477015078130821291417187305759, 10.59867488777300404521478170834, 11.37925311020530026779770712966

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