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Properties

Label 2-315-63.25-c1-0-28
Degree $2$
Conductor $315$
Sign $-0.500 + 0.865i$
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  + 1.19·2-s + (−0.773 − 1.54i)3-s − 0.564·4-s + (0.5 − 0.866i)5-s + (−0.926 − 1.85i)6-s + (0.433 − 2.60i)7-s − 3.07·8-s + (−1.80 + 2.39i)9-s + (0.599 − 1.03i)10-s + (−0.0568 − 0.0985i)11-s + (0.436 + 0.875i)12-s + (−1.79 − 3.11i)13-s + (0.519 − 3.12i)14-s + (−1.72 − 0.105i)15-s − 2.55·16-s + (1.61 − 2.80i)17-s + ⋯
L(s)  = 1  + 0.847·2-s + (−0.446 − 0.894i)3-s − 0.282·4-s + (0.223 − 0.387i)5-s + (−0.378 − 0.758i)6-s + (0.163 − 0.986i)7-s − 1.08·8-s + (−0.601 + 0.799i)9-s + (0.189 − 0.328i)10-s + (−0.0171 − 0.0297i)11-s + (0.126 + 0.252i)12-s + (−0.498 − 0.863i)13-s + (0.138 − 0.835i)14-s + (−0.446 − 0.0271i)15-s − 0.637·16-s + (0.392 − 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.500 + 0.865i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657825 - 1.14008i\)
\(L(\frac12)\) \(\approx\) \(0.657825 - 1.14008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.773 + 1.54i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.433 + 2.60i)T \)
good2 \( 1 - 1.19T + 2T^{2} \)
11 \( 1 + (0.0568 + 0.0985i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (-0.300 + 0.519i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.57 + 2.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.81T + 31T^{2} \)
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 \( 1 - 8.66T + 47T^{2} \)
53 \( 1 + (5.54 - 9.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 4.63T + 67T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 + (-6.73 + 11.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
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

−11.74001974599283771308576919280, −10.57294017257937965114401319627, −9.569140778134801663214653205709, −8.230251801644059239459832590649, −7.42640059270593071471099033236, −6.20860868368675772498449297275, −5.30545030384900378205644591324, −4.40635120872365468444466259853, −2.86564881926535018997842108205, −0.799818850705314061467883286697, 2.70908438316722075767921654894, 3.94177818744866924712289247664, 4.97957428386066052145893049395, 5.72250398350568053839918880757, 6.67303110652679777946383552516, 8.489287426490092727427391980415, 9.284649913447495878646581147077, 10.06599298254980383305209697602, 11.26454739581362137976156875305, 12.01466633016426898997509532803

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