↑

Properties

Label 2-315-63.25-c1-0-8
Degree $2$
Conductor $315$
Sign $0.666 - 0.745i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.699·2-s + (−1.56 − 0.735i)3-s − 1.51·4-s + (−0.5 + 0.866i)5-s + (−1.09 − 0.514i)6-s + (2.62 + 0.363i)7-s − 2.45·8-s + (1.91 + 2.30i)9-s + (−0.349 + 0.605i)10-s + (1.91 + 3.32i)11-s + (2.37 + 1.11i)12-s + (2.28 + 3.94i)13-s + (1.83 + 0.254i)14-s + (1.42 − 0.990i)15-s + 1.30·16-s + (2.93 − 5.07i)17-s + ⋯
L(s)  = 1  + 0.494·2-s + (−0.905 − 0.424i)3-s − 0.755·4-s + (−0.223 + 0.387i)5-s + (−0.447 − 0.209i)6-s + (0.990 + 0.137i)7-s − 0.867·8-s + (0.639 + 0.768i)9-s + (−0.110 + 0.191i)10-s + (0.578 + 1.00i)11-s + (0.684 + 0.320i)12-s + (0.632 + 1.09i)13-s + (0.489 + 0.0679i)14-s + (0.366 − 0.255i)15-s + 0.326·16-s + (0.711 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.666 - 0.745i$
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: \((2,\ 315,\ (\ :1/2),\ 0.666 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.960990 + 0.430104i\)
\(L(\frac12)\) \(\approx\) \(0.960990 + 0.430104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 + 0.735i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.62 - 0.363i)T \)
good2 \( 1 - 0.699T + 2T^{2} \)
11 \( 1 + (-1.91 - 3.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.28 - 3.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.93 + 5.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.386 - 0.670i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.45 - 5.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.95 - 6.84i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.03T + 31T^{2} \)
37 \( 1 + (-4.23 - 7.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.60 + 2.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.19 + 7.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.33T + 47T^{2} \)
53 \( 1 + (-1.32 + 2.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.73T + 59T^{2} \)
61 \( 1 - 7.01T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 + (-7.00 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 + (-0.442 + 0.766i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.950 - 1.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.07 - 5.32i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.77842407937390968424897004975, −11.30263130898533581397724041489, −9.954975838145391987826168245134, −9.051606169776000216636550570459, −7.72301567103494296622245396362, −6.90261358189973210917067026957, −5.62677052544539516906964246879, −4.83021822414840747111035500955, −3.82232001490540514613629069570, −1.63209374558849093541708198147, 0.847194652158945330251247776454, 3.70075840642798861326185232385, 4.36411521002239459489437253457, 5.60557615777360946096960517300, 6.02000530589293428566019640716, 7.953943539280060451168649835866, 8.611963446442851575567660482688, 9.787424406102478178709295350163, 10.82891389228295942310152234647, 11.50080977135157957943891355666

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