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Properties

Label 2-315-63.20-c1-0-29
Degree $2$
Conductor $315$
Sign $-0.900 + 0.433i$
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  + (−0.334 − 0.193i)2-s + (0.454 − 1.67i)3-s + (−0.925 − 1.60i)4-s + (−0.5 − 0.866i)5-s + (−0.475 + 0.471i)6-s + (1.06 − 2.41i)7-s + 1.48i·8-s + (−2.58 − 1.51i)9-s + 0.386i·10-s + (3.67 + 2.12i)11-s + (−3.09 + 0.818i)12-s + (−3.15 + 1.81i)13-s + (−0.826 + 0.603i)14-s + (−1.67 + 0.442i)15-s + (−1.56 + 2.70i)16-s − 2.70·17-s + ⋯
L(s)  = 1  + (−0.236 − 0.136i)2-s + (0.262 − 0.964i)3-s + (−0.462 − 0.801i)4-s + (−0.223 − 0.387i)5-s + (−0.194 + 0.192i)6-s + (0.404 − 0.914i)7-s + 0.526i·8-s + (−0.862 − 0.506i)9-s + 0.122i·10-s + (1.10 + 0.639i)11-s + (−0.894 + 0.236i)12-s + (−0.873 + 0.504i)13-s + (−0.220 + 0.161i)14-s + (−0.432 + 0.114i)15-s + (−0.390 + 0.676i)16-s − 0.655·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.900 + 0.433i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.900 + 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.220829 - 0.967310i\)
\(L(\frac12)\) \(\approx\) \(0.220829 - 0.967310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.454 + 1.67i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.06 + 2.41i)T \)
good2 \( 1 + (0.334 + 0.193i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-3.67 - 2.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.15 - 1.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + 5.17iT - 19T^{2} \)
23 \( 1 + (-3.29 + 1.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.67 - 2.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.652 - 0.376i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + (-0.143 - 0.248i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.08 + 8.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.82 + 8.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.71iT - 53T^{2} \)
59 \( 1 + (-4.44 - 7.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.5 - 6.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.74 + 8.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 + (1.05 - 1.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.95 - 12.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.28T + 89T^{2} \)
97 \( 1 + (-11.7 - 6.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.35112779687678218183421933987, −10.31394207980797232482543562960, −9.127026638425079536012877914192, −8.681118816766126423006246569074, −7.19829272829572167516681198093, −6.74687011930204114160167591595, −5.11752045739379950446776567396, −4.18094666750895280779895343429, −2.08327917557613099264494431157, −0.789747690875144649718252726580, 2.76018082078182653107105383028, 3.79108586260571843874291398905, 4.85198305050722227808119400158, 6.11752444854807853729801427829, 7.59866808475705362739602941036, 8.469975331232928404155949210735, 9.127750744021934190695709307225, 9.970153404774372885417087678247, 11.21249769190491161351162443109, 11.88436121056140745719707712131

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