↑

Properties

Label 2-845-65.9-c1-0-10
Degree $2$
Conductor $845$
Sign $0.795 + 0.605i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (−1.05 − 0.607i)2-s + (−1.13 − 0.655i)3-s + (−0.262 − 0.455i)4-s + (−2.21 − 0.311i)5-s + (0.796 + 1.37i)6-s + (−2.51 + 1.45i)7-s + 3.06i·8-s + (−0.640 − 1.10i)9-s + (2.13 + 1.67i)10-s + (−0.107 + 0.185i)11-s + 0.688i·12-s + 3.52·14-s + (2.31 + 1.80i)15-s + (1.33 − 2.31i)16-s + (−5.56 + 3.21i)17-s + 1.55i·18-s + ⋯
L(s)  = 1  + (−0.743 − 0.429i)2-s + (−0.655 − 0.378i)3-s + (−0.131 − 0.227i)4-s + (−0.990 − 0.139i)5-s + (0.324 + 0.562i)6-s + (−0.950 + 0.548i)7-s + 1.08i·8-s + (−0.213 − 0.369i)9-s + (0.676 + 0.528i)10-s + (−0.0323 + 0.0559i)11-s + 0.198i·12-s + 0.942·14-s + (0.596 + 0.466i)15-s + (0.334 − 0.578i)16-s + (−1.35 + 0.779i)17-s + 0.366i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.795 + 0.605i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.795 + 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250036 - 0.0842991i\)
\(L(\frac12)\) \(\approx\) \(0.250036 - 0.0842991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.21 + 0.311i)T \)
13 \( 1 \)
good2 \( 1 + (1.05 + 0.607i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.13 + 0.655i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.51 - 1.45i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.107 - 0.185i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (5.56 - 3.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.06 + 2.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.35 + 7.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.59T + 31T^{2} \)
37 \( 1 + (-1.97 - 1.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.52 - 2.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.50 + 3.18i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.09iT - 47T^{2} \)
53 \( 1 + 6.23iT - 53T^{2} \)
59 \( 1 + (4.63 + 8.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.140 - 0.243i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.72 + 3.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 9.52iT - 83T^{2} \)
89 \( 1 + (2.80 - 4.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.6 + 9.02i)T + (48.5 - 84.0i)T^{2} \)
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

−10.12002811137750831302440164527, −9.234365177590976914487209909664, −8.602222888376272907749225543568, −7.75868339718171278041173730468, −6.46749105903807033461258165029, −5.99813204998004691725773432826, −4.75808246979997480413751794783, −3.58427982551577876298284103749, −2.19005631974084675147205596277, −0.52872043866317283647491159351, 0.39356071344400451870826040298, 2.98799257048815582448882406230, 4.01011349236760662114523053673, 4.81155411091492944779462068471, 6.20218967571219308128328603762, 7.09303177679665815262009437667, 7.57941466686852871485512135814, 8.670819025713077453637839229126, 9.295595170877840512904944610548, 10.30634902882935659097337681188

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