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Properties

Label 2-315-63.5-c1-0-12
Degree $2$
Conductor $315$
Sign $0.360 - 0.932i$
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  + 0.103i·2-s + (0.570 + 1.63i)3-s + 1.98·4-s + (−0.5 − 0.866i)5-s + (−0.170 + 0.0593i)6-s + (−0.107 + 2.64i)7-s + 0.414i·8-s + (−2.34 + 1.86i)9-s + (0.0900 − 0.0519i)10-s + (2.53 + 1.46i)11-s + (1.13 + 3.25i)12-s + (−2.40 − 1.38i)13-s + (−0.274 − 0.0111i)14-s + (1.13 − 1.31i)15-s + 3.93·16-s + (−3.62 − 6.27i)17-s + ⋯
L(s)  = 1  + 0.0735i·2-s + (0.329 + 0.944i)3-s + 0.994·4-s + (−0.223 − 0.387i)5-s + (−0.0694 + 0.0242i)6-s + (−0.0406 + 0.999i)7-s + 0.146i·8-s + (−0.782 + 0.622i)9-s + (0.0284 − 0.0164i)10-s + (0.764 + 0.441i)11-s + (0.327 + 0.939i)12-s + (−0.666 − 0.385i)13-s + (−0.0734 − 0.00298i)14-s + (0.292 − 0.338i)15-s + 0.983·16-s + (−0.878 − 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.360 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38884 + 0.952454i\)
\(L(\frac12)\) \(\approx\) \(1.38884 + 0.952454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.570 - 1.63i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.107 - 2.64i)T \)
good2 \( 1 - 0.103iT - 2T^{2} \)
11 \( 1 + (-2.53 - 1.46i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.40 + 1.38i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.62 + 6.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.40 - 3.69i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.159 - 0.0920i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.84 - 1.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.83iT - 31T^{2} \)
37 \( 1 + (1.26 - 2.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.223 - 0.387i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.588T + 47T^{2} \)
53 \( 1 + (-6.46 + 3.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.14T + 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 - 3.10T + 67T^{2} \)
71 \( 1 + 5.62iT - 71T^{2} \)
73 \( 1 + (-11.3 + 6.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 9.11T + 79T^{2} \)
83 \( 1 + (4.90 + 8.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.98 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.69 - 0.981i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.81085155401508072488684850286, −11.04017683554018617916364193403, −9.725479219514421232581518005754, −9.279362687384110570849749874424, −8.071855055612384351859735485083, −7.11236067225952994098720728215, −5.72281791438110163597711965374, −4.89117707494569299925327401017, −3.37466396286022682519476396735, −2.25726893695447259834468098256, 1.36226462395758843646434855060, 2.79822339748056085178469644384, 3.95659429329633728383446549772, 5.95983090312283009337427789132, 6.92064294761144443301094077939, 7.30595100243465873489908551386, 8.408414669140571808877801622151, 9.659538912090408199995081102662, 10.88322921103013702260410904247, 11.44340493590021523462101522010

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