↑

Properties

Label 2-833-7.2-c1-0-22
Degree $2$
Conductor $833$
Sign $0.968 - 0.250i$
Analytic cond. $6.65153$
Root an. cond. $2.57905$
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 + (0.500 + 0.866i)4-s + (−1 + 1.73i)5-s + 3·8-s + (1.5 − 2.59i)9-s + (0.999 + 1.73i)10-s + 2·13-s + (0.500 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−1.5 − 2.59i)18-s + (−2 + 3.46i)19-s − 2·20-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s + (1 − 1.73i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.447 + 0.774i)5-s + 1.06·8-s + (0.5 − 0.866i)9-s + (0.316 + 0.547i)10-s + 0.554·13-s + (0.125 − 0.216i)16-s + (0.121 + 0.210i)17-s + (−0.353 − 0.612i)18-s + (−0.458 + 0.794i)19-s − 0.447·20-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s + (0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(833\)    =    \(7^{2} \cdot 17\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(6.65153\)
Root analytic conductor: \(2.57905\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{833} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 833,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05893 + 0.262377i\)
\(L(\frac12)\) \(\approx\) \(2.05893 + 0.262377i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−10.53064312420803616035304111638, −9.624095842787528466424901517514, −8.449217985742049287391560548159, −7.62204770543795494265207749363, −6.84221173028454487133407565717, −6.01068772037633017845342788940, −4.43601302403092627094106619637, −3.66517940780322195078532386187, −2.96012938507344229839176282744, −1.51503257309667408895772474316, 1.05630170281451120834976790512, 2.48031671858594864192237311718, 4.40222696606554414379261330314, 4.64590786529415646612354416146, 5.81700235391606697061146107554, 6.63281393937014306622019542268, 7.61766362888151390270908750930, 8.256987634786088433829642449513, 9.233667381650447385374823033574, 10.36293870541928097194567298036

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