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Properties

Label 2-315-63.58-c1-0-24
Degree $2$
Conductor $315$
Sign $-0.992 + 0.118i$
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  − 2.69·2-s + (−0.273 − 1.71i)3-s + 5.27·4-s + (−0.5 − 0.866i)5-s + (0.739 + 4.61i)6-s + (−0.230 − 2.63i)7-s − 8.84·8-s + (−2.84 + 0.937i)9-s + (1.34 + 2.33i)10-s + (1.96 − 3.39i)11-s + (−1.44 − 9.02i)12-s + (0.993 − 1.72i)13-s + (0.621 + 7.11i)14-s + (−1.34 + 1.09i)15-s + 13.2·16-s + (2.23 + 3.87i)17-s + ⋯
L(s)  = 1  − 1.90·2-s + (−0.158 − 0.987i)3-s + 2.63·4-s + (−0.223 − 0.387i)5-s + (0.301 + 1.88i)6-s + (−0.0870 − 0.996i)7-s − 3.12·8-s + (−0.949 + 0.312i)9-s + (0.426 + 0.738i)10-s + (0.591 − 1.02i)11-s + (−0.417 − 2.60i)12-s + (0.275 − 0.477i)13-s + (0.166 + 1.90i)14-s + (−0.347 + 0.282i)15-s + 3.32·16-s + (0.542 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0219021 - 0.367323i\)
\(L(\frac12)\) \(\approx\) \(0.0219021 - 0.367323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.273 + 1.71i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.230 + 2.63i)T \)
good2 \( 1 + 2.69T + 2T^{2} \)
11 \( 1 + (-1.96 + 3.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.993 + 1.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.23 - 3.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0804 + 0.139i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.11 + 5.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.384 + 0.666i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.03T + 31T^{2} \)
37 \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.42 + 2.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.53 + 2.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 + (-5.50 - 9.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.36T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 6.78T + 71T^{2} \)
73 \( 1 + (7.20 + 12.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + (2.40 + 4.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.16 + 8.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.51 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.90930730873008155748904385983, −10.39632783446476925716887537309, −9.064190819887623598298891725239, −8.281474873570761203977130876142, −7.71078192627860446608436100076, −6.69161882844412374249902385714, −5.91363113251945723194106926367, −3.38944624911126737964837870166, −1.60641261031618686758352851754, −0.49855290843048262855124325686, 2.06733527334993366017561457867, 3.48566917037327524795900560776, 5.43863703822010583397148997739, 6.59265005481768522741751245515, 7.58024777030577828590993774551, 8.685665029740145282383205815449, 9.535756157736922337939343099258, 9.767266059598061115357332933977, 11.02873472750237105101860665913, 11.60377196709169583613436534007

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