↑

Properties

Label 2-315-63.59-c1-0-16
Degree $2$
Conductor $315$
Sign $0.429 - 0.903i$
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  + (2.36 + 1.36i)2-s + (−1.43 − 0.972i)3-s + (2.72 + 4.71i)4-s + 5-s + (−2.05 − 4.25i)6-s + (1.49 − 2.18i)7-s + 9.39i·8-s + (1.10 + 2.78i)9-s + (2.36 + 1.36i)10-s − 0.284i·11-s + (0.686 − 9.40i)12-s + (−2.89 − 1.67i)13-s + (6.50 − 3.12i)14-s + (−1.43 − 0.972i)15-s + (−7.36 + 12.7i)16-s + (−2.65 + 4.60i)17-s + ⋯
L(s)  = 1  + (1.67 + 0.964i)2-s + (−0.827 − 0.561i)3-s + (1.36 + 2.35i)4-s + 0.447·5-s + (−0.840 − 1.73i)6-s + (0.564 − 0.825i)7-s + 3.32i·8-s + (0.368 + 0.929i)9-s + (0.747 + 0.431i)10-s − 0.0856i·11-s + (0.198 − 2.71i)12-s + (−0.803 − 0.464i)13-s + (1.73 − 0.835i)14-s + (−0.369 − 0.251i)15-s + (−1.84 + 3.19i)16-s + (−0.644 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.429 - 0.903i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.429 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29477 + 1.45004i\)
\(L(\frac12)\) \(\approx\) \(2.29477 + 1.45004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.43 + 0.972i)T \)
5 \( 1 - T \)
7 \( 1 + (-1.49 + 2.18i)T \)
good2 \( 1 + (-2.36 - 1.36i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 0.284iT - 11T^{2} \)
13 \( 1 + (2.89 + 1.67i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.65 - 4.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.743 + 0.429i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.91iT - 23T^{2} \)
29 \( 1 + (1.98 - 1.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.992 + 0.572i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.53 + 7.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.15 + 5.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.223 - 0.387i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.00 - 5.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.483 - 0.279i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.59 - 7.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.96 + 4.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.802 + 1.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.30iT - 71T^{2} \)
73 \( 1 + (-3.55 - 2.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.29 - 3.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.29 - 3.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.97 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.69 - 2.13i)T + (48.5 - 84.0i)T^{2} \)
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

−12.38099152853238987442451405315, −11.14319169562857404214487801616, −10.52975137959220985719363807106, −8.390922674079897521334310043264, −7.46009621634910682016724409294, −6.74553714326905446119326158204, −5.85334076556701079411364419403, −4.95791438282671847171051991407, −4.10220099444776340233116746294, −2.29229727531155416597606695042, 1.80446712173957139655593614522, 3.15732067414506575423630720891, 4.61779014889822603370077987497, 5.13819697479564786421303705061, 5.98643975028228683164058245542, 7.04013670688488181187009149981, 9.349195624178982263639828871181, 9.912096628721747823254241297437, 11.05988459004767788120024874778, 11.70095659205636394150160504484

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