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Properties

Label 2-315-63.4-c1-0-29
Degree $2$
Conductor $315$
Sign $-0.665 + 0.746i$
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 − 2.21i)2-s + (1.68 + 0.390i)3-s + (−2.27 − 3.94i)4-s − 5-s + (3.02 − 3.24i)6-s + (−1.62 − 2.08i)7-s − 6.53·8-s + (2.69 + 1.31i)9-s + (−1.28 + 2.21i)10-s + 3.81·11-s + (−2.30 − 7.54i)12-s + (−0.798 + 1.38i)13-s + (−6.70 + 0.939i)14-s + (−1.68 − 0.390i)15-s + (−3.81 + 6.60i)16-s + (−1.84 + 3.18i)17-s + ⋯
L(s)  = 1  + (0.905 − 1.56i)2-s + (0.974 + 0.225i)3-s + (−1.13 − 1.97i)4-s − 0.447·5-s + (1.23 − 1.32i)6-s + (−0.615 − 0.788i)7-s − 2.31·8-s + (0.898 + 0.439i)9-s + (−0.404 + 0.701i)10-s + 1.14·11-s + (−0.664 − 2.17i)12-s + (−0.221 + 0.383i)13-s + (−1.79 + 0.251i)14-s + (−0.435 − 0.100i)15-s + (−0.953 + 1.65i)16-s + (−0.446 + 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.665 + 0.746i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.665 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961948 - 2.14768i\)
\(L(\frac12)\) \(\approx\) \(0.961948 - 2.14768i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.390i)T \)
5 \( 1 + T \)
7 \( 1 + (1.62 + 2.08i)T \)
good2 \( 1 + (-1.28 + 2.21i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 3.81T + 11T^{2} \)
13 \( 1 + (0.798 - 1.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.84 - 3.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.68 - 2.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.904T + 23T^{2} \)
29 \( 1 + (4.97 + 8.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.27 - 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.90 - 6.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.38 + 11.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.849 - 1.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.84 - 8.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.61 - 2.80i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.12 + 1.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.12 - 8.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.83 - 3.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (1.47 - 2.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.98 + 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.67 + 4.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.998 + 1.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.25 + 9.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.41147614150026561778146231794, −10.46137976373386007620103262138, −9.716904831022584284011964370945, −8.975643654782073488479100990103, −7.55394894779971196394334571804, −6.19677564438991214333942817289, −4.35456974190788240377966723983, −3.98820428215259844360047452328, −2.96726066372338429581602675515, −1.48930355737182636782713573886, 2.92794835570477849829731286370, 3.91580143337182028200717196820, 5.07122232860482753082814572427, 6.37283893032018243950661527579, 7.06854079627329698062703661670, 7.918409553901739351665991585088, 8.992165129904449412975119384697, 9.398390391770275390471878283977, 11.50780589438043040044281571993, 12.54268853442332615667243026166

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