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Properties

Label 2-315-63.16-c1-0-22
Degree $2$
Conductor $315$
Sign $-0.0679 + 0.997i$
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  + (−0.599 − 1.03i)2-s + (1.72 + 0.105i)3-s + (0.282 − 0.488i)4-s − 5-s + (−0.926 − 1.85i)6-s + (2.04 − 1.68i)7-s − 3.07·8-s + (2.97 + 0.363i)9-s + (0.599 + 1.03i)10-s + 0.113·11-s + (0.539 − 0.815i)12-s + (−1.79 − 3.11i)13-s + (−2.96 − 1.11i)14-s + (−1.72 − 0.105i)15-s + (1.27 + 2.21i)16-s + (1.61 + 2.80i)17-s + ⋯
L(s)  = 1  + (−0.423 − 0.733i)2-s + (0.998 + 0.0607i)3-s + (0.141 − 0.244i)4-s − 0.447·5-s + (−0.378 − 0.758i)6-s + (0.772 − 0.635i)7-s − 1.08·8-s + (0.992 + 0.121i)9-s + (0.189 + 0.328i)10-s + 0.0342·11-s + (0.155 − 0.235i)12-s + (−0.498 − 0.863i)13-s + (−0.793 − 0.297i)14-s + (−0.446 − 0.0271i)15-s + (0.318 + 0.552i)16-s + (0.392 + 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0679 + 0.997i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.0679 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01184 - 1.08312i\)
\(L(\frac12)\) \(\approx\) \(1.01184 - 1.08312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.105i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.04 + 1.68i)T \)
good2 \( 1 + (0.599 + 1.03i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 - 0.113T + 11T^{2} \)
13 \( 1 + (1.79 + 3.11i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.61 - 2.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.951 + 1.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.600T + 23T^{2} \)
29 \( 1 + (-1.57 + 2.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.90 - 5.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.60 - 2.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.74 - 8.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.780 + 1.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.33 + 7.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.54 + 9.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.16 - 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.79 - 11.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.31 - 4.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 + (-6.73 - 11.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.73 - 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0205 + 0.0356i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.22 - 2.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.36 - 4.09i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.22239039555498230357452393857, −10.37473230524587085870253782763, −9.779213697050492667504328689974, −8.576650503376225582076232079910, −7.904083961842492976787233277763, −6.84980221953502724334692998141, −5.19133173793623943852466865451, −3.84413838599039478469602125580, −2.69266672525907055371606518617, −1.27309315621906587062790882541, 2.18404477399715309351513382655, 3.47630531944269983477634796384, 4.85426334444474849375361407544, 6.36911291623089795704183254722, 7.58603003000692779377003669701, 7.83934205670891634136492774220, 9.014574761169348290826145376680, 9.428515762893329665711580669171, 11.05202978298332970620430272281, 12.06084948136253519374864457704

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