↑

Properties

Label 2-315-63.47-c1-0-28
Degree $2$
Conductor $315$
Sign $0.456 + 0.889i$
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  + (2.11 − 1.22i)2-s + (1.73 + 0.0669i)3-s + (1.98 − 3.44i)4-s − 5-s + (3.74 − 1.97i)6-s + (−2.50 + 0.855i)7-s − 4.82i·8-s + (2.99 + 0.231i)9-s + (−2.11 + 1.22i)10-s + 4.38i·11-s + (3.67 − 5.82i)12-s + (−1.54 + 0.891i)13-s + (−4.25 + 4.87i)14-s + (−1.73 − 0.0669i)15-s + (−1.92 − 3.33i)16-s + (−3.92 − 6.80i)17-s + ⋯
L(s)  = 1  + (1.49 − 0.864i)2-s + (0.999 + 0.0386i)3-s + (0.993 − 1.72i)4-s − 0.447·5-s + (1.52 − 0.805i)6-s + (−0.946 + 0.323i)7-s − 1.70i·8-s + (0.997 + 0.0772i)9-s + (−0.669 + 0.386i)10-s + 1.32i·11-s + (1.05 − 1.68i)12-s + (−0.428 + 0.247i)13-s + (−1.13 + 1.30i)14-s + (−0.446 − 0.0172i)15-s + (−0.481 − 0.833i)16-s + (−0.953 − 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.77214 - 1.69293i\)
\(L(\frac12)\) \(\approx\) \(2.77214 - 1.69293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0669i)T \)
5 \( 1 + T \)
7 \( 1 + (2.50 - 0.855i)T \)
good2 \( 1 + (-2.11 + 1.22i)T + (1 - 1.73i)T^{2} \)
11 \( 1 - 4.38iT - 11T^{2} \)
13 \( 1 + (1.54 - 0.891i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.92 + 6.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.33 + 1.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.26iT - 23T^{2} \)
29 \( 1 + (-0.771 - 0.445i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.13 - 3.53i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.58 + 4.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.78 + 3.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.68 + 4.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.299 - 0.519i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.9 - 6.87i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.46 - 7.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.76 + 2.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.24 + 5.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.343iT - 71T^{2} \)
73 \( 1 + (3.38 - 1.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.43 + 5.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.09 + 5.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.69 + 2.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.3 - 5.95i)T + (48.5 + 84.0i)T^{2} \)
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

−11.95331629178156794852028402086, −10.73162202887906049762082123933, −9.742409367455533605573830462920, −9.018042666198041100944141654096, −7.34320692236531492082457358224, −6.58194556767852293496417618369, −4.88149442243364054613202426137, −4.23101980287652384920851762024, −2.97995909966877473023840751658, −2.23312334451951803028792419467, 2.80629362443689929967248310711, 3.71784693291991584993166520649, 4.46924724804856299601086540444, 6.17394290209356023708380413002, 6.62076413937860676584660334392, 7.969519341974464088299193613443, 8.452867965459903239045162522271, 9.935384723182136621172299035457, 11.09672523610938612793354843255, 12.46320969422923458533684083975

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