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Properties

Label 2-315-63.47-c1-0-4
Degree $2$
Conductor $315$
Sign $0.0210 - 0.999i$
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  + (−1.67 + 0.967i)2-s + (−1.71 + 0.243i)3-s + (0.873 − 1.51i)4-s − 5-s + (2.63 − 2.06i)6-s + (2.28 + 1.32i)7-s − 0.490i·8-s + (2.88 − 0.834i)9-s + (1.67 − 0.967i)10-s − 4.75i·11-s + (−1.12 + 2.80i)12-s + (−1.44 + 0.834i)13-s + (−5.12 − 0.00860i)14-s + (1.71 − 0.243i)15-s + (2.22 + 3.84i)16-s + (0.551 + 0.954i)17-s + ⋯
L(s)  = 1  + (−1.18 + 0.684i)2-s + (−0.990 + 0.140i)3-s + (0.436 − 0.756i)4-s − 0.447·5-s + (1.07 − 0.844i)6-s + (0.865 + 0.501i)7-s − 0.173i·8-s + (0.960 − 0.278i)9-s + (0.530 − 0.306i)10-s − 1.43i·11-s + (−0.326 + 0.810i)12-s + (−0.400 + 0.231i)13-s + (−1.36 − 0.00230i)14-s + (0.442 − 0.0628i)15-s + (0.555 + 0.961i)16-s + (0.133 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0210 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0210 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0210 - 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0210 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344530 + 0.337353i\)
\(L(\frac12)\) \(\approx\) \(0.344530 + 0.337353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.243i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.28 - 1.32i)T \)
good2 \( 1 + (1.67 - 0.967i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + 4.75iT - 11T^{2} \)
13 \( 1 + (1.44 - 0.834i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.551 - 0.954i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.85 - 2.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.57iT - 23T^{2} \)
29 \( 1 + (-2.66 - 1.54i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.43 + 1.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.755 + 1.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.80 - 6.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0488 + 0.0845i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.75 - 4.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.1 + 6.42i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.68 - 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.30 + 5.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.53 + 4.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 + (-3.66 + 2.11i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.10 - 5.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.60 - 4.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.15 - 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.33 + 0.768i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.52415770473536593104896304249, −11.01806847237715830923695745222, −9.871707435666080993956262098362, −9.009592306435131105388987868156, −7.992151207830570396331436454361, −7.34600912726393874406841765529, −6.06879915950968144927546124859, −5.30749679948576884225689867456, −3.74420310088901065950753097740, −1.13290145065819504619963582228, 0.74997953179731221082846782888, 2.21056314477695269082559323551, 4.41537341450126185328036743049, 5.23727750696712210732341332778, 7.06313679194778527996639557619, 7.56082983777373800408122292493, 8.705304362498599805582423826005, 9.940486623627043902953742148542, 10.44375627247451382172786792226, 11.28975498532267310487495686902

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