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Properties

Label 2-315-63.59-c1-0-11
Degree $2$
Conductor $315$
Sign $0.848 + 0.528i$
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  + (−0.0900 − 0.0519i)2-s + (−1.13 + 1.31i)3-s + (−0.994 − 1.72i)4-s − 5-s + (0.170 − 0.0593i)6-s + (2.34 + 1.22i)7-s + 0.414i·8-s + (−0.442 − 2.96i)9-s + (0.0900 + 0.0519i)10-s − 2.92i·11-s + (3.38 + 0.643i)12-s + (2.40 + 1.38i)13-s + (−0.147 − 0.232i)14-s + (1.13 − 1.31i)15-s + (−1.96 + 3.40i)16-s + (3.62 − 6.27i)17-s + ⋯
L(s)  = 1  + (−0.0636 − 0.0367i)2-s + (−0.652 + 0.757i)3-s + (−0.497 − 0.861i)4-s − 0.447·5-s + (0.0694 − 0.0242i)6-s + (0.885 + 0.464i)7-s + 0.146i·8-s + (−0.147 − 0.989i)9-s + (0.0284 + 0.0164i)10-s − 0.882i·11-s + (0.977 + 0.185i)12-s + (0.666 + 0.385i)13-s + (−0.0393 − 0.0621i)14-s + (0.292 − 0.338i)15-s + (−0.491 + 0.852i)16-s + (0.878 − 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.848 + 0.528i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.848 + 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.890874 - 0.254922i\)
\(L(\frac12)\) \(\approx\) \(0.890874 - 0.254922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.13 - 1.31i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.34 - 1.22i)T \)
good2 \( 1 + (0.0900 + 0.0519i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 2.92iT - 11T^{2} \)
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 \( 1 + 0.184iT - 23T^{2} \)
29 \( 1 + (1.84 - 1.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.45 + 1.41i)T + (15.5 - 26.8i)T^{2} \)
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 \( 1 + (-0.294 + 0.509i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.46 + 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.07 + 7.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.9 + 6.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.55 + 2.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.62iT - 71T^{2} \)
73 \( 1 + (-11.3 - 6.58i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.55 + 7.89i)T + (-39.5 - 68.4i)T^{2} \)
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

−11.34406833346360157459422122091, −10.90542341439197624423445841549, −9.581186587362943384107476477606, −9.103773299909077408252426571451, −7.918350280696026849000552937174, −6.40743870994676595622705517382, −5.28348754626526524076688993561, −4.84387624307250591649386537030, −3.34943308234785245104719903815, −0.909936438371119753116329203847, 1.42637461615620429516173099190, 3.52044568016704945276414050380, 4.65855298612068147181854248188, 5.79686566443797184715361705767, 7.29161518588367924074840419195, 7.79310049780525315870239759987, 8.496808032325751482593811047786, 10.05805237255105564364352098514, 10.96549109503378534247497957109, 12.10888403997675085396802355524

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