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Properties

Label 2-315-315.157-c1-0-30
Degree $2$
Conductor $315$
Sign $0.979 - 0.199i$
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  + (1.86 − 0.5i)2-s + (1.5 + 0.866i)3-s + (1.5 − 0.866i)4-s + (−1 + 2i)5-s + (3.23 + 0.866i)6-s + (0.5 − 2.59i)7-s + (−0.366 + 0.366i)8-s + (1.5 + 2.59i)9-s + (−0.866 + 4.23i)10-s − 2·11-s + 3·12-s + (5.09 − 1.36i)13-s + (−0.366 − 5.09i)14-s + (−3.23 + 2.13i)15-s + (−2.23 + 3.86i)16-s + (−2 − 7.46i)17-s + ⋯
L(s)  = 1  + (1.31 − 0.353i)2-s + (0.866 + 0.499i)3-s + (0.750 − 0.433i)4-s + (−0.447 + 0.894i)5-s + (1.31 + 0.353i)6-s + (0.188 − 0.981i)7-s + (−0.129 + 0.129i)8-s + (0.5 + 0.866i)9-s + (−0.273 + 1.33i)10-s − 0.603·11-s + 0.866·12-s + (1.41 − 0.378i)13-s + (−0.0978 − 1.36i)14-s + (−0.834 + 0.550i)15-s + (−0.558 + 0.966i)16-s + (−0.485 − 1.81i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.979 - 0.199i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.979 - 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90181 + 0.292742i\)
\(L(\frac12)\) \(\approx\) \(2.90181 + 0.292742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (1 - 2i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 + (-1.86 + 0.5i)T + (1.73 - i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-5.09 + 1.36i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2 + 7.46i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.36 + 2.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.46 - 2.46i)T - 23iT^{2} \)
29 \( 1 + (3.40 - 1.96i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.26 - 0.732i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.169 + 0.633i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.23 - 0.866i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.96 + 7.33i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1 + 3.73i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-7.56 - 13.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.53 - 1.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.63 - 6.09i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (-3.46 + 0.928i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.92 - 2.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.03 - 11.3i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (7.46 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.5 - 3.90i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.47160826795345119265353076945, −11.03332420410684306036419920374, −10.16465408308672066548084922874, −8.839501426442382188316729364861, −7.74444111927351896539776234571, −6.83157815093248238595824809838, −5.32676003476244896249595330050, −4.20060564429181003069209707598, −3.49342752989715642964795902295, −2.53234337901932275657105376221, 1.95375124650096116528043270208, 3.58939512979782220026773467010, 4.33257234090569640457857215130, 5.74328101702387394860156968493, 6.37872242163070681176785742978, 7.976222126115902399742427476820, 8.514039117233975052435344875040, 9.410419191596320644666890270175, 11.07969399653084833086535258637, 12.22785230743399038085953595576

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