↑

Properties

Label 2-845-169.118-c1-0-22
Degree $2$
Conductor $845$
Sign $0.949 + 0.312i$
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  + (−1.79 − 1.59i)2-s + (−0.481 + 1.26i)3-s + (0.453 + 3.73i)4-s + (0.970 + 0.239i)5-s + (2.88 − 1.51i)6-s + (0.683 − 0.989i)7-s + (2.40 − 3.49i)8-s + (0.864 + 0.766i)9-s + (−1.36 − 1.97i)10-s + (0.0189 − 0.0168i)11-s + (−4.96 − 1.22i)12-s + (3.44 + 1.05i)13-s + (−2.80 + 0.691i)14-s + (−0.771 + 1.11i)15-s + (−2.57 + 0.634i)16-s + (4.20 − 6.08i)17-s + ⋯
L(s)  = 1  + (−1.27 − 1.12i)2-s + (−0.278 + 0.733i)3-s + (0.226 + 1.86i)4-s + (0.434 + 0.107i)5-s + (1.17 − 0.618i)6-s + (0.258 − 0.374i)7-s + (0.851 − 1.23i)8-s + (0.288 + 0.255i)9-s + (−0.431 − 0.624i)10-s + (0.00572 − 0.00507i)11-s + (−1.43 − 0.353i)12-s + (0.955 + 0.293i)13-s + (−0.749 + 0.184i)14-s + (−0.199 + 0.288i)15-s + (−0.643 + 0.158i)16-s + (1.01 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (456, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.949 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855244 - 0.136955i\)
\(L(\frac12)\) \(\approx\) \(0.855244 - 0.136955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (-3.44 - 1.05i)T \)
good2 \( 1 + (1.79 + 1.59i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (0.481 - 1.26i)T + (-2.24 - 1.98i)T^{2} \)
7 \( 1 + (-0.683 + 0.989i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (-0.0189 + 0.0168i)T + (1.32 - 10.9i)T^{2} \)
17 \( 1 + (-4.20 + 6.08i)T + (-6.02 - 15.8i)T^{2} \)
19 \( 1 + 5.59T + 19T^{2} \)
23 \( 1 + 0.577T + 23T^{2} \)
29 \( 1 + (-3.85 - 3.41i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (5.22 - 2.74i)T + (17.6 - 25.5i)T^{2} \)
37 \( 1 + (-6.84 + 3.59i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (-0.184 + 0.486i)T + (-30.6 - 27.1i)T^{2} \)
43 \( 1 + (-6.48 - 3.40i)T + (24.4 + 35.3i)T^{2} \)
47 \( 1 + (0.173 - 1.43i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (-4.83 + 7.00i)T + (-18.7 - 49.5i)T^{2} \)
59 \( 1 + (-4.47 - 1.10i)T + (52.2 + 27.4i)T^{2} \)
61 \( 1 + (-4.42 - 6.41i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (-1.36 + 11.2i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (-4.83 + 12.7i)T + (-53.1 - 47.0i)T^{2} \)
73 \( 1 + (11.2 - 9.96i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (0.791 - 6.51i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (-3.04 - 8.03i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + (4.21 - 1.03i)T + (85.8 - 45.0i)T^{2} \)
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

−10.19117865014490485001626450473, −9.512480396728392536482092827181, −8.864642830297904574603948727470, −7.896163078298815593110683256871, −7.03575402719404185398022961498, −5.67045912439464043999489563587, −4.49683080637898393080371191315, −3.49291474604380003878339003542, −2.27530009729365791297351890006, −1.04535846723457781227216904621, 0.913837740671875932060423334152, 1.94162081085239883674784620031, 3.99849237763554156842362003922, 5.73738255563887743698902326733, 6.02734536173074074654343835088, 6.81903406213218576586109534112, 7.83928042799132650695945446710, 8.363769654675373969351669688228, 9.118345789339488321345958072975, 10.10999913152022636265843238898

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