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Properties

Label 2-315-63.25-c1-0-6
Degree $2$
Conductor $315$
Sign $-0.468 - 0.883i$
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  + 1.50·2-s + (−1.11 + 1.32i)3-s + 0.251·4-s + (−0.5 + 0.866i)5-s + (−1.67 + 1.98i)6-s + (0.0793 + 2.64i)7-s − 2.62·8-s + (−0.512 − 2.95i)9-s + (−0.750 + 1.29i)10-s + (1.41 + 2.45i)11-s + (−0.280 + 0.333i)12-s + (0.0336 + 0.0582i)13-s + (0.119 + 3.96i)14-s + (−0.590 − 1.62i)15-s − 4.44·16-s + (−3.48 + 6.02i)17-s + ⋯
L(s)  = 1  + 1.06·2-s + (−0.643 + 0.765i)3-s + 0.125·4-s + (−0.223 + 0.387i)5-s + (−0.683 + 0.811i)6-s + (0.0299 + 0.999i)7-s − 0.927·8-s + (−0.170 − 0.985i)9-s + (−0.237 + 0.410i)10-s + (0.426 + 0.739i)11-s + (−0.0810 + 0.0963i)12-s + (0.00932 + 0.0161i)13-s + (0.0318 + 1.06i)14-s + (−0.152 − 0.420i)15-s − 1.11·16-s + (−0.844 + 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.468 - 0.883i$
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: \((2,\ 315,\ (\ :1/2),\ -0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692253 + 1.15051i\)
\(L(\frac12)\) \(\approx\) \(0.692253 + 1.15051i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.11 - 1.32i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.0793 - 2.64i)T \)
good2 \( 1 - 1.50T + 2T^{2} \)
11 \( 1 + (-1.41 - 2.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0336 - 0.0582i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.48 - 6.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.667 - 1.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.61 + 2.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.51 + 7.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.47T + 31T^{2} \)
37 \( 1 + (-1.37 - 2.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.36 - 4.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 + (6.77 - 11.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.87T + 59T^{2} \)
61 \( 1 - 3.79T + 61T^{2} \)
67 \( 1 - 3.65T + 67T^{2} \)
71 \( 1 + 8.89T + 71T^{2} \)
73 \( 1 + (-2.58 + 4.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + (5.40 - 9.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.42 + 5.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.45 + 2.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.13612814205267919107956687788, −11.30687645125806886810001361192, −10.22374796727997906611684072525, −9.276809445872550827766939474178, −8.337364933122922399615675945918, −6.43162368296034074876255756301, −6.03444012320849546332474369336, −4.70573053765096295937829259213, −4.11360703812274190633919600722, −2.72101717233941296876112620190, 0.77775012193153803655126115997, 3.02847517043391761311858914543, 4.45177594681950809832197333806, 5.15200591201820987736413918259, 6.38339154349102226912869681839, 7.13275864212943650267977402780, 8.363624985451437445222441693928, 9.461038948820587530185833839056, 10.94744303098902218840198980868, 11.55669670516671748252782215680

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