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Properties

Label 2-315-63.59-c1-0-5
Degree $2$
Conductor $315$
Sign $0.603 - 0.797i$
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  + (−1.07 − 0.618i)2-s + (1.48 + 0.888i)3-s + (−0.235 − 0.407i)4-s − 5-s + (−1.04 − 1.87i)6-s + (−2.10 + 1.60i)7-s + 3.05i·8-s + (1.42 + 2.64i)9-s + (1.07 + 0.618i)10-s + 1.76i·11-s + (0.0120 − 0.815i)12-s + (5.15 + 2.97i)13-s + (3.24 − 0.415i)14-s + (−1.48 − 0.888i)15-s + (1.41 − 2.45i)16-s + (−2.04 + 3.53i)17-s + ⋯
L(s)  = 1  + (−0.757 − 0.437i)2-s + (0.858 + 0.512i)3-s + (−0.117 − 0.203i)4-s − 0.447·5-s + (−0.425 − 0.763i)6-s + (−0.795 + 0.605i)7-s + 1.08i·8-s + (0.474 + 0.880i)9-s + (0.338 + 0.195i)10-s + 0.533i·11-s + (0.00347 − 0.235i)12-s + (1.43 + 0.826i)13-s + (0.867 − 0.110i)14-s + (−0.383 − 0.229i)15-s + (0.354 − 0.614i)16-s + (−0.495 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.603 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808076 + 0.401638i\)
\(L(\frac12)\) \(\approx\) \(0.808076 + 0.401638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.48 - 0.888i)T \)
5 \( 1 + T \)
7 \( 1 + (2.10 - 1.60i)T \)
good2 \( 1 + (1.07 + 0.618i)T + (1 + 1.73i)T^{2} \)
11 \( 1 - 1.76iT - 11T^{2} \)
13 \( 1 + (-5.15 - 2.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.04 - 3.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.71 + 2.72i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.17iT - 23T^{2} \)
29 \( 1 + (7.08 - 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.211 + 0.122i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.29 + 3.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.718 - 1.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.63 - 6.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.00762 - 0.0132i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.05 - 3.49i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.414 - 0.717i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.3 + 6.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.85 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + (4.15 + 2.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.03 - 5.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.03 + 3.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.64 + 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.65 + 2.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.41156945715578162729052957945, −10.74045414708896331363951988337, −9.708519052477710614024274241673, −9.016773609130882932805031170452, −8.551865934377766692839180168335, −7.26919571174692767173292042336, −5.86078693587764049799965477817, −4.46079267945502320352238158605, −3.26052877076256638766370637480, −1.83909552953285676744282626426, 0.811437183604747475868977824390, 3.27750886758558652428622742423, 3.81405520769792082847203377316, 6.00919023143862337818245219454, 7.13322871416893517834306745214, 7.72489458046219801583724665493, 8.602361055847694834467733585738, 9.345667825284821421386855565943, 10.25524964459347136139184625493, 11.54017093981691690873072092188

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