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Properties

Label 2-315-105.2-c1-0-7
Degree $2$
Conductor $315$
Sign $0.920 - 0.390i$
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  + (0.0362 + 0.135i)2-s + (1.71 − 0.990i)4-s + (1.87 + 1.21i)5-s + (0.702 + 2.55i)7-s + (0.394 + 0.394i)8-s + (−0.0969 + 0.297i)10-s + (−4.85 + 2.80i)11-s + (2.57 − 2.57i)13-s + (−0.319 + 0.187i)14-s + (1.94 − 3.36i)16-s + (−0.269 − 0.0721i)17-s + (−4.44 − 2.56i)19-s + (4.42 + 0.234i)20-s + (−0.554 − 0.554i)22-s + (4.46 − 1.19i)23-s + ⋯
L(s)  = 1  + (0.0256 + 0.0956i)2-s + (0.857 − 0.495i)4-s + (0.838 + 0.545i)5-s + (0.265 + 0.964i)7-s + (0.139 + 0.139i)8-s + (−0.0306 + 0.0941i)10-s + (−1.46 + 0.844i)11-s + (0.714 − 0.714i)13-s + (−0.0854 + 0.0501i)14-s + (0.485 − 0.840i)16-s + (−0.0653 − 0.0175i)17-s + (−1.01 − 0.588i)19-s + (0.988 + 0.0523i)20-s + (−0.118 − 0.118i)22-s + (0.931 − 0.249i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71591 + 0.348829i\)
\(L(\frac12)\) \(\approx\) \(1.71591 + 0.348829i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 - 1.21i)T \)
7 \( 1 + (-0.702 - 2.55i)T \)
good2 \( 1 + (-0.0362 - 0.135i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (4.85 - 2.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.57 + 2.57i)T - 13iT^{2} \)
17 \( 1 + (0.269 + 0.0721i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.44 + 2.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.46 + 1.19i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.91T + 29T^{2} \)
31 \( 1 + (1.09 + 1.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.35 + 1.96i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.73iT - 41T^{2} \)
43 \( 1 + (5.90 - 5.90i)T - 43iT^{2} \)
47 \( 1 + (2.91 + 10.8i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.417 + 1.55i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.62 - 2.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.06 - 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.85 - 6.92i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.16iT - 71T^{2} \)
73 \( 1 + (7.92 + 2.12i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.500 + 0.289i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.12 + 3.12i)T + 83iT^{2} \)
89 \( 1 + (6.29 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.25 + 8.25i)T + 97iT^{2} \)
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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

−11.47872651089381806207953500722, −10.70604248752086857059747799359, −10.12816243382434620014699556984, −8.966620042360806256211885575691, −7.77186496498184690639826241302, −6.74390369476600756292778124165, −5.77183053525589843536312089095, −5.09205381466646992687901417498, −2.80029125427660813704907064409, −2.05722120407031838424633990049, 1.57281659211189503655257720231, 3.01916471241978165109365491807, 4.42892654555304048271553233251, 5.77333283013015075398523605396, 6.69552367244864688806658573801, 7.87579591791300042171474584374, 8.576756488755795940208070254746, 9.942363780603454752707450740281, 10.85191723699433811874194643165, 11.30555154922633904483126583451

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