↑

Properties

Label 2-845-65.2-c1-0-58
Degree $2$
Conductor $845$
Sign $-0.384 + 0.922i$
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  + (0.5 + 0.866i)2-s + (−1.36 − 0.366i)3-s + (0.500 − 0.866i)4-s + (1 − 2i)5-s + (−0.366 − 1.36i)6-s + (−1.73 − i)7-s + 3·8-s + (−0.866 − 0.5i)9-s + (2.23 − 0.133i)10-s + (0.366 − 1.36i)11-s + (−1 + 0.999i)12-s − 1.99i·14-s + (−2.09 + 2.36i)15-s + (0.500 + 0.866i)16-s + (0.366 + 1.36i)17-s − i·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.788 − 0.211i)3-s + (0.250 − 0.433i)4-s + (0.447 − 0.894i)5-s + (−0.149 − 0.557i)6-s + (−0.654 − 0.377i)7-s + 1.06·8-s + (−0.288 − 0.166i)9-s + (0.705 − 0.0423i)10-s + (0.110 − 0.411i)11-s + (−0.288 + 0.288i)12-s − 0.534i·14-s + (−0.541 + 0.610i)15-s + (0.125 + 0.216i)16-s + (0.0887 + 0.331i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.384 + 0.922i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.384 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626960 - 0.940713i\)
\(L(\frac12)\) \(\approx\) \(0.626960 - 0.940713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1 + 2i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.36 + 0.366i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.366 + 1.36i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.366 - 1.36i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.83 - 1.83i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.09 + 4.09i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 - 5i)T - 31iT^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.56 + 2.56i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.36 + 0.366i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (-2.56 - 9.56i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.366 + 1.36i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-6.83 - 1.83i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.19549272061727955806387828851, −8.955859630241914972372358941897, −8.262033229595846567634401329591, −6.80368157951640867007143410890, −6.44580752645922514953039390207, −5.60980605495080133617974476459, −4.94957849590935818683758767804, −3.76841766050824419956630416040, −1.91208787449550532397909569638, −0.50834165305357770876765121683, 2.09547701036749397807307736023, 2.91226321948930522806055986296, 3.98050418044936732793021391095, 5.12875216718643415235672610039, 6.13399703354736778658056938385, 6.81205952455584408522521634303, 7.73808921676559757555037762598, 8.968744263262889397160377606994, 9.982290964246733707617549114306, 10.63736560339051093358910780671

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