↑

Properties

Label 2-315-63.58-c1-0-2
Degree $2$
Conductor $315$
Sign $0.207 - 0.978i$
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  − 1.42·2-s + (−1.59 + 0.674i)3-s + 0.0328·4-s + (−0.5 − 0.866i)5-s + (2.27 − 0.962i)6-s + (−1.98 − 1.75i)7-s + 2.80·8-s + (2.08 − 2.15i)9-s + (0.712 + 1.23i)10-s + (1.61 − 2.79i)11-s + (−0.0524 + 0.0221i)12-s + (−2.89 + 5.02i)13-s + (2.82 + 2.49i)14-s + (1.38 + 1.04i)15-s − 4.06·16-s + (1.62 + 2.81i)17-s + ⋯
L(s)  = 1  − 1.00·2-s + (−0.920 + 0.389i)3-s + 0.0164·4-s + (−0.223 − 0.387i)5-s + (0.928 − 0.392i)6-s + (−0.749 − 0.662i)7-s + 0.991·8-s + (0.696 − 0.717i)9-s + (0.225 + 0.390i)10-s + (0.487 − 0.844i)11-s + (−0.0151 + 0.00640i)12-s + (−0.804 + 1.39i)13-s + (0.755 + 0.667i)14-s + (0.356 + 0.269i)15-s − 1.01·16-s + (0.393 + 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.207 - 0.978i$
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: \((2,\ 315,\ (\ :1/2),\ 0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253301 + 0.205207i\)
\(L(\frac12)\) \(\approx\) \(0.253301 + 0.205207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 - 0.674i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.98 + 1.75i)T \)
good2 \( 1 + 1.42T + 2T^{2} \)
11 \( 1 + (-1.61 + 2.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.89 - 5.02i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.38 - 4.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.878 + 1.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.61 - 4.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.41T + 31T^{2} \)
37 \( 1 + (3.46 - 6.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.27 - 2.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.147 - 0.255i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + (-1.20 - 2.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.623T + 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 - 0.494T + 67T^{2} \)
71 \( 1 - 5.06T + 71T^{2} \)
73 \( 1 + (5.79 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 + (-0.961 - 1.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.95 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.37 - 7.57i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.74789065117228727199442965949, −10.57282504777183069036937006518, −10.06267868912466944201177203294, −9.195334572858964706599842322359, −8.298079771712100875846583420530, −7.01971548444847185477779377658, −6.20511178643726527303212797671, −4.66648604212732563431552835733, −3.86809722138883805841129046661, −1.14537217330065109894081068115, 0.45158930851085784293669187018, 2.49726562758120061455417161648, 4.49716598640235113769583801050, 5.62502087926350031173441459651, 6.89523095716589780489048950294, 7.51914732633549874275524315774, 8.661478373471491789281182363696, 9.942654689053296843204232707404, 10.13015421232011235107905494532, 11.39852826055275519837279368024

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