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Properties

Label 2-315-63.58-c1-0-22
Degree $2$
Conductor $315$
Sign $0.744 - 0.667i$
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  + 2.34·2-s + (0.308 + 1.70i)3-s + 3.47·4-s + (0.5 + 0.866i)5-s + (0.723 + 3.98i)6-s + (−1.61 − 2.09i)7-s + 3.45·8-s + (−2.80 + 1.05i)9-s + (1.17 + 2.02i)10-s + (1.62 − 2.82i)11-s + (1.07 + 5.92i)12-s + (0.549 − 0.951i)13-s + (−3.78 − 4.89i)14-s + (−1.32 + 1.11i)15-s + 1.13·16-s + (0.763 + 1.32i)17-s + ⋯
L(s)  = 1  + 1.65·2-s + (0.178 + 0.983i)3-s + 1.73·4-s + (0.223 + 0.387i)5-s + (0.295 + 1.62i)6-s + (−0.611 − 0.791i)7-s + 1.22·8-s + (−0.936 + 0.351i)9-s + (0.369 + 0.640i)10-s + (0.490 − 0.850i)11-s + (0.310 + 1.71i)12-s + (0.152 − 0.263i)13-s + (−1.01 − 1.30i)14-s + (−0.341 + 0.289i)15-s + 0.282·16-s + (0.185 + 0.320i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90620 + 1.11252i\)
\(L(\frac12)\) \(\approx\) \(2.90620 + 1.11252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.308 - 1.70i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.61 + 2.09i)T \)
good2 \( 1 - 2.34T + 2T^{2} \)
11 \( 1 + (-1.62 + 2.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.549 + 0.951i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.763 - 1.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.06 - 5.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.15 + 3.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.48 + 2.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + (-4.94 + 8.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.222 - 0.385i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.84 - 4.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.23T + 47T^{2} \)
53 \( 1 + (-5.48 - 9.50i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.55T + 59T^{2} \)
61 \( 1 + 7.48T + 61T^{2} \)
67 \( 1 - 2.13T + 67T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 + (-4.96 - 8.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.86T + 79T^{2} \)
83 \( 1 + (8.19 + 14.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.440 + 0.762i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.41 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.89115032411487869442505987461, −10.81838358921453201364823956551, −10.35647191081421646477080316543, −9.111329981614141370881610199830, −7.75347361509803940350142901527, −6.20264503844566098692713528001, −5.90113551517277273233992030635, −4.30684868846413443936930767652, −3.77381542485520971802395392008, −2.73073008976320831145870061286, 2.02082856261776247498936327811, 3.07293358700836135611712773592, 4.50086943362785911296738405211, 5.62264196058336763300099105820, 6.45123377021186537714330121268, 7.18556157881545241537731231134, 8.672635574971764815064909292540, 9.573799004755335681140824405396, 11.27497312537201747100011416529, 12.09377883459835543859662618972

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