↑

Properties

Label 2-315-63.47-c1-0-27
Degree $2$
Conductor $315$
Sign $0.0549 + 0.998i$
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  + (1.48 − 0.855i)2-s + (1.02 − 1.39i)3-s + (0.463 − 0.802i)4-s − 5-s + (0.324 − 2.94i)6-s + (1.20 − 2.35i)7-s + 1.83i·8-s + (−0.897 − 2.86i)9-s + (−1.48 + 0.855i)10-s − 0.574i·11-s + (−0.645 − 1.46i)12-s + (0.130 − 0.0752i)13-s + (−0.236 − 4.52i)14-s + (−1.02 + 1.39i)15-s + (2.49 + 4.32i)16-s + (0.586 + 1.01i)17-s + ⋯
L(s)  = 1  + (1.04 − 0.604i)2-s + (0.591 − 0.805i)3-s + (0.231 − 0.401i)4-s − 0.447·5-s + (0.132 − 1.20i)6-s + (0.454 − 0.890i)7-s + 0.649i·8-s + (−0.299 − 0.954i)9-s + (−0.468 + 0.270i)10-s − 0.173i·11-s + (−0.186 − 0.424i)12-s + (0.0361 − 0.0208i)13-s + (−0.0632 − 1.20i)14-s + (−0.264 + 0.360i)15-s + (0.624 + 1.08i)16-s + (0.142 + 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0549 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0549 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0549 + 0.998i$
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: \((2,\ 315,\ (\ :1/2),\ 0.0549 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76339 - 1.66909i\)
\(L(\frac12)\) \(\approx\) \(1.76339 - 1.66909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.02 + 1.39i)T \)
5 \( 1 + T \)
7 \( 1 + (-1.20 + 2.35i)T \)
good2 \( 1 + (-1.48 + 0.855i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + 0.574iT - 11T^{2} \)
13 \( 1 + (-0.130 + 0.0752i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.586 - 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.00148 - 0.000858i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.46iT - 23T^{2} \)
29 \( 1 + (-5.81 - 3.35i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.74 + 3.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.718 - 1.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.37 - 9.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.00 - 5.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.22 + 7.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.60 - 5.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.13 + 8.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.36 + 3.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.10 + 7.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.0708iT - 71T^{2} \)
73 \( 1 + (-10.2 + 5.91i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.65 + 2.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.47 - 7.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.699 - 1.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.9 + 8.61i)T + (48.5 + 84.0i)T^{2} \)
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

−11.54342667458117774918742024731, −11.05341998015146813191443550234, −9.623168741661168355477293421248, −8.262396110053582765553941214936, −7.73104820291838988945916055666, −6.54365560414375294842231436187, −5.15368453849953874402417171048, −3.93014141477295293916070668858, −3.13846273929237566048939049790, −1.56477395712581382476789621000, 2.63725525771927336215953674637, 3.94017163848113431193514888578, 4.80926552402586713138493301217, 5.62549310401212094494248326472, 6.90842309089639374390353085216, 8.117258776837015411837837604718, 8.934496636365363602454292145077, 9.980635965850268165795524623138, 10.99995898088621968271928069486, 12.14375267907544887195549355928

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