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Properties

Label 2-315-35.27-c1-0-12
Degree $2$
Conductor $315$
Sign $0.326 - 0.945i$
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 + (2.58 − 0.581i)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 + (0.975 − 0.219i)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

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

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.326 - 0.945i$
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: \((2,\ 315,\ (\ :1/2),\ 0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08951 + 1.48963i\)
\(L(\frac12)\) \(\approx\) \(2.08951 + 1.48963i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.58 + 0.581i)T \)
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

−12.44550620652980315402755932101, −10.86251003688420206743625329738, −10.10851637423159379171328482494, −8.568847692519159129022402468505, −7.88583741487788696754675606465, −6.80101704423884812625169301091, −5.64942346642628363652043351894, −5.10785996193417288669520267594, −4.09427928447877804697660892320, −2.19525511880401949247953721663, 2.09378454373959694315309691927, 2.59226495478940619391045598478, 4.45051500596848503880084145313, 5.04046967552420222045112657525, 6.20719188114977512150411302082, 7.50828274124002056125600379794, 8.946233987376226098496903983275, 10.03131326430229035383768563274, 10.82362165852681487670001681702, 11.49619208903978159941767640298

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