Properties

Label 2-315-63.16-c1-0-19
Degree $2$
Conductor $315$
Sign $-0.871 + 0.490i$
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  + (−1.29 − 2.23i)2-s + (1.72 − 0.177i)3-s + (−2.34 + 4.05i)4-s + 5-s + (−2.62 − 3.62i)6-s + (−2.13 − 1.55i)7-s + 6.94·8-s + (2.93 − 0.612i)9-s + (−1.29 − 2.23i)10-s + 2.47·11-s + (−3.31 + 7.40i)12-s + (−3.19 − 5.52i)13-s + (−0.718 + 6.80i)14-s + (1.72 − 0.177i)15-s + (−4.29 − 7.43i)16-s + (−2.57 − 4.45i)17-s + ⋯
L(s)  = 1  + (−0.914 − 1.58i)2-s + (0.994 − 0.102i)3-s + (−1.17 + 2.02i)4-s + 0.447·5-s + (−1.07 − 1.48i)6-s + (−0.808 − 0.588i)7-s + 2.45·8-s + (0.978 − 0.204i)9-s + (−0.408 − 0.708i)10-s + 0.744·11-s + (−0.957 + 2.13i)12-s + (−0.885 − 1.53i)13-s + (−0.191 + 1.81i)14-s + (0.444 − 0.0458i)15-s + (−1.07 − 1.85i)16-s + (−0.623 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.871 + 0.490i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.871 + 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264814 - 1.01028i\)
\(L(\frac12)\) \(\approx\) \(0.264814 - 1.01028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.177i)T \)
5 \( 1 - T \)
7 \( 1 + (2.13 + 1.55i)T \)
good2 \( 1 + (1.29 + 2.23i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 + (3.19 + 5.52i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.57 + 4.45i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.825 + 1.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 + (-0.321 + 0.557i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.273 - 0.474i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.30 - 5.72i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.42 - 7.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.75 - 6.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.817 - 1.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.66 - 8.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.44 + 7.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.157 - 0.272i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.593 + 1.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.21T + 71T^{2} \)
73 \( 1 + (-6.30 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.14 + 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.21 + 5.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.44 - 2.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.53 + 9.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.04199781235541659727868287726, −10.05478662105255065397985147300, −9.615543662020809238091045611936, −8.892007810751274972611097673514, −7.79843100735885999652829396207, −6.87254339943428359100315273350, −4.62409026814658935573205881801, −3.23086961078991455579027310810, −2.67306194978508943261679319135, −0.979560271119353689735002071453, 1.99417160528886978841990614605, 4.05800729670641447103988415673, 5.46378152776107744657938531839, 6.72796546557248867585384419342, 7.04484241989644068065594029677, 8.482855032676743090745244987383, 9.117207525325591579360224639941, 9.512058852995104026452357211319, 10.49336042726015830806024234365, 12.24268782236793409036272650000

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