↑

Properties

Label 2-845-845.389-c1-0-75
Degree $2$
Conductor $845$
Sign $0.958 + 0.284i$
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 + 0.938i)2-s + (0.713 + 0.270i)3-s + (0.000188 + 0.00155i)4-s + (1.74 − 1.40i)5-s + (0.501 + 0.956i)6-s + (−0.0120 + 0.0175i)7-s + (1.60 − 2.32i)8-s + (−1.80 − 1.60i)9-s + (3.16 + 0.146i)10-s + (−1.52 − 1.72i)11-s + (−0.000285 + 0.00115i)12-s + (−2.65 − 2.44i)13-s + (−0.0292 + 0.00720i)14-s + (1.62 − 0.530i)15-s + (3.88 − 0.958i)16-s + (6.72 + 4.64i)17-s + ⋯
L(s)  = 1  + (0.748 + 0.663i)2-s + (0.412 + 0.156i)3-s + (9.41e−5 + 0.000775i)4-s + (0.778 − 0.627i)5-s + (0.204 + 0.390i)6-s + (−0.00457 + 0.00662i)7-s + (0.567 − 0.822i)8-s + (−0.603 − 0.534i)9-s + (0.999 + 0.0464i)10-s + (−0.460 − 0.520i)11-s + (−8.24e−5 + 0.000334i)12-s + (−0.735 − 0.677i)13-s + (−0.00781 + 0.00192i)14-s + (0.418 − 0.137i)15-s + (0.971 − 0.239i)16-s + (1.63 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.958 + 0.284i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79224 - 0.405698i\)
\(L(\frac12)\) \(\approx\) \(2.79224 - 0.405698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.74 + 1.40i)T \)
13 \( 1 + (2.65 + 2.44i)T \)
good2 \( 1 + (-1.05 - 0.938i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (-0.713 - 0.270i)T + (2.24 + 1.98i)T^{2} \)
7 \( 1 + (0.0120 - 0.0175i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (1.52 + 1.72i)T + (-1.32 + 10.9i)T^{2} \)
17 \( 1 + (-6.72 - 4.64i)T + (6.02 + 15.8i)T^{2} \)
19 \( 1 - 2.31iT - 19T^{2} \)
23 \( 1 + 1.05iT - 23T^{2} \)
29 \( 1 + (-2.64 - 2.34i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (1.71 + 3.26i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (-2.00 + 1.05i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (0.217 + 0.0825i)T + (30.6 + 27.1i)T^{2} \)
43 \( 1 + (2.92 - 5.56i)T + (-24.4 - 35.3i)T^{2} \)
47 \( 1 + (1.01 - 8.39i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (-2.24 - 1.54i)T + (18.7 + 49.5i)T^{2} \)
59 \( 1 + (1.97 - 8.02i)T + (-52.2 - 27.4i)T^{2} \)
61 \( 1 + (-4.20 - 6.09i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (-0.971 + 8.00i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (-9.73 - 3.69i)T + (53.1 + 47.0i)T^{2} \)
73 \( 1 + (-2.36 + 2.09i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (-0.662 + 5.45i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (1.79 + 4.74i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 - 17.1iT - 89T^{2} \)
97 \( 1 + (11.1 - 2.74i)T + (85.8 - 45.0i)T^{2} \)
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

−10.01770225471950960305263642595, −9.404178459620316310001566622972, −8.287767501849211429798544972056, −7.68395505544449100285260334930, −6.22063590061041038408504294654, −5.78868795928321276207970533900, −5.07601719219434103854088692794, −3.90353306011447044279626197360, −2.82318815632807299696332772878, −1.11635480361878942908596888281, 2.01036535692163668296893275977, 2.67096752026867909777334637059, 3.50747239139404105038235677979, 5.02329342076198624297855451230, 5.37561265456709859673489910406, 6.92892426873049942720200094152, 7.57126545761118231544126117641, 8.528235920062742279288806781898, 9.670698939678416137102243814990, 10.23109296482666744873981512984

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