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Properties

Label 2-315-63.4-c1-0-13
Degree $2$
Conductor $315$
Sign $0.153 - 0.988i$
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  + (−1.37 + 2.38i)2-s + (−1.12 + 1.31i)3-s + (−2.80 − 4.85i)4-s + 5-s + (−1.60 − 4.49i)6-s + (2.15 − 1.53i)7-s + 9.92·8-s + (−0.481 − 2.96i)9-s + (−1.37 + 2.38i)10-s + 2.28·11-s + (9.54 + 1.74i)12-s + (1.37 − 2.38i)13-s + (0.705 + 7.26i)14-s + (−1.12 + 1.31i)15-s + (−8.08 + 14.0i)16-s + (1.01 − 1.75i)17-s + ⋯
L(s)  = 1  + (−0.974 + 1.68i)2-s + (−0.647 + 0.761i)3-s + (−1.40 − 2.42i)4-s + 0.447·5-s + (−0.654 − 1.83i)6-s + (0.813 − 0.581i)7-s + 3.51·8-s + (−0.160 − 0.987i)9-s + (−0.435 + 0.755i)10-s + 0.687·11-s + (2.75 + 0.504i)12-s + (0.382 − 0.662i)13-s + (0.188 + 1.94i)14-s + (−0.289 + 0.340i)15-s + (−2.02 + 3.50i)16-s + (0.245 − 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.153 - 0.988i$
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: \((2,\ 315,\ (\ :1/2),\ 0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534179 + 0.457423i\)
\(L(\frac12)\) \(\approx\) \(0.534179 + 0.457423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.12 - 1.31i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.15 + 1.53i)T \)
good2 \( 1 + (1.37 - 2.38i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 2.28T + 11T^{2} \)
13 \( 1 + (-1.37 + 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.01 + 1.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.03 + 5.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.99T + 23T^{2} \)
29 \( 1 + (2.78 + 4.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.166 - 0.288i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0601 - 0.104i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.237 - 0.411i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.43 - 5.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.60 - 6.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.25 - 7.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.32 - 7.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.64 + 9.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.39 - 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (-0.0952 + 0.165i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.77 + 9.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.98 + 6.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.05 - 1.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.42 + 5.93i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.26469237927236051852555779100, −10.68913487967277613957717182024, −9.695220279293505887409170710517, −9.057543037841648171712248829172, −8.056149221726389100306665330106, −6.96216977297205767629509563479, −6.13878041613771510668393589635, −5.17213002861036483815047322898, −4.35566002016162168288950829830, −0.917363632620533165891332400405, 1.41038524947947228149067386272, 2.13342510375473735893918369870, 3.87062293009292050829034280533, 5.28232259582407988800921479864, 6.81557480877472151504935137940, 8.122322409970517816609918568264, 8.710715724651796296884921093946, 9.761352406856173274436305428826, 10.83406058244981227300408376880, 11.30564394085691319179141812056

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