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Properties

Label 2-315-15.8-c1-0-7
Degree $2$
Conductor $315$
Sign $0.830 - 0.557i$
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  + (1.28 + 1.28i)2-s + 1.28i·4-s + (0.565 − 2.16i)5-s + (0.707 − 0.707i)7-s + (0.918 − 0.918i)8-s + (3.49 − 2.04i)10-s + 4.14i·11-s + (4.57 + 4.57i)13-s + 1.81·14-s + 4.92·16-s + (−5.27 − 5.27i)17-s − 3.06i·19-s + (2.77 + 0.725i)20-s + (−5.30 + 5.30i)22-s + (−3.82 + 3.82i)23-s + ⋯
L(s)  = 1  + (0.905 + 0.905i)2-s + 0.641i·4-s + (0.252 − 0.967i)5-s + (0.267 − 0.267i)7-s + (0.324 − 0.324i)8-s + (1.10 − 0.647i)10-s + 1.24i·11-s + (1.26 + 1.26i)13-s + 0.484·14-s + 1.23·16-s + (−1.27 − 1.27i)17-s − 0.702i·19-s + (0.620 + 0.162i)20-s + (−1.13 + 1.13i)22-s + (−0.797 + 0.797i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.830 - 0.557i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.830 - 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12456 + 0.647371i\)
\(L(\frac12)\) \(\approx\) \(2.12456 + 0.647371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.565 + 2.16i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.28 - 1.28i)T + 2iT^{2} \)
11 \( 1 - 4.14iT - 11T^{2} \)
13 \( 1 + (-4.57 - 4.57i)T + 13iT^{2} \)
17 \( 1 + (5.27 + 5.27i)T + 17iT^{2} \)
19 \( 1 + 3.06iT - 19T^{2} \)
23 \( 1 + (3.82 - 3.82i)T - 23iT^{2} \)
29 \( 1 + 5.24T + 29T^{2} \)
31 \( 1 + 2.42T + 31T^{2} \)
37 \( 1 + (0.834 - 0.834i)T - 37iT^{2} \)
41 \( 1 + 4.19iT - 41T^{2} \)
43 \( 1 + (-3.71 - 3.71i)T + 43iT^{2} \)
47 \( 1 + (-3.14 - 3.14i)T + 47iT^{2} \)
53 \( 1 + (4.79 - 4.79i)T - 53iT^{2} \)
59 \( 1 + 1.97T + 59T^{2} \)
61 \( 1 - 1.88T + 61T^{2} \)
67 \( 1 + (10.7 - 10.7i)T - 67iT^{2} \)
71 \( 1 + 9.76iT - 71T^{2} \)
73 \( 1 + (-0.978 - 0.978i)T + 73iT^{2} \)
79 \( 1 - 4.09iT - 79T^{2} \)
83 \( 1 + (-3.68 + 3.68i)T - 83iT^{2} \)
89 \( 1 + 7.39T + 89T^{2} \)
97 \( 1 + (3.07 - 3.07i)T - 97iT^{2} \)
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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

−11.96320983347755402046941054052, −11.02320456173078305690572709988, −9.578076847903522976661060822774, −8.965898965925892841989438321526, −7.56280985260195866481161726375, −6.80383880669533432677697705132, −5.73401774142051687231928941911, −4.62990131737119458315530464733, −4.15763579166485795153789806696, −1.73213572941184700107499969409, 1.96386158091869062717575421094, 3.24543163931492185495462868481, 3.96518657271871246730791506533, 5.67740850382860577428410208474, 6.19041471946510195586456057662, 7.916512005174910719478932628092, 8.641644434358633487994245953817, 10.35551638211741291249697015478, 10.86434108657524460827877813173, 11.37049891804343977325556488199

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