↑

Properties

Label 2-315-9.4-c1-0-10
Degree $2$
Conductor $315$
Sign $-0.997 - 0.0700i$
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  + (−1.32 − 2.30i)2-s + (−0.649 + 1.60i)3-s + (−2.53 + 4.38i)4-s + (−0.5 + 0.866i)5-s + (4.55 − 0.639i)6-s + (0.5 + 0.866i)7-s + 8.13·8-s + (−2.15 − 2.08i)9-s + 2.65·10-s + (−2.87 − 4.97i)11-s + (−5.39 − 6.90i)12-s + (−0.0700 + 0.121i)13-s + (1.32 − 2.30i)14-s + (−1.06 − 1.36i)15-s + (−5.74 − 9.95i)16-s + 1.53·17-s + ⋯
L(s)  = 1  + (−0.939 − 1.62i)2-s + (−0.374 + 0.927i)3-s + (−1.26 + 2.19i)4-s + (−0.223 + 0.387i)5-s + (1.86 − 0.261i)6-s + (0.188 + 0.327i)7-s + 2.87·8-s + (−0.719 − 0.694i)9-s + 0.840·10-s + (−0.866 − 1.50i)11-s + (−1.55 − 1.99i)12-s + (−0.0194 + 0.0336i)13-s + (0.355 − 0.615i)14-s + (−0.275 − 0.352i)15-s + (−1.43 − 2.48i)16-s + 0.372·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.997 - 0.0700i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.997 - 0.0700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00764202 + 0.217844i\)
\(L(\frac12)\) \(\approx\) \(0.00764202 + 0.217844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.649 - 1.60i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.32 + 2.30i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.87 + 4.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0700 - 0.121i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.53T + 17T^{2} \)
19 \( 1 + 6.87T + 19T^{2} \)
23 \( 1 + (-2.07 + 3.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.59 + 6.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.23 + 5.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.03T + 37T^{2} \)
41 \( 1 + (5.23 - 9.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.28 + 2.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0840 - 0.145i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + (-4.25 + 7.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.635 - 1.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.249 - 0.431i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.34T + 71T^{2} \)
73 \( 1 - 2.16T + 73T^{2} \)
79 \( 1 + (-0.300 - 0.519i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.37 + 5.84i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (-2.58 - 4.47i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.14432783587385125640656403087, −10.41553169696762752161339024386, −9.659402464597705855555945114946, −8.542062137907015284708385640821, −8.112154934271479814812275694953, −6.11826886614575547090594661780, −4.61681881445366624318197091423, −3.49199726447473763361330931579, −2.54596026367671654381809706839, −0.22429190384785843888969560176, 1.63048048261139953603470622943, 4.74067528806508307934940187011, 5.46551998698935602008818526416, 6.74884944490330702768485449484, 7.30046901184447757080007393454, 8.080403266721501992770141253645, 8.882578105306467437259749399144, 10.09642696655067589034980290719, 10.85914274756078703118446627123, 12.39347471643188947785728404928

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