↑

Properties

Label 2-34e2-1156.851-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.984 + 0.173i$
Analytic cond. $0.576919$
Root an. cond. $0.759551$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.932 − 0.361i)2-s + (0.739 − 0.673i)4-s + (0.397 + 0.798i)5-s + (0.445 − 0.895i)8-s + (−0.273 + 0.961i)9-s + (0.658 + 0.600i)10-s + (−0.243 + 0.489i)13-s + (0.0922 − 0.995i)16-s + (0.0922 − 0.995i)17-s + (0.0922 + 0.995i)18-s + (0.831 + 0.322i)20-s + (0.123 − 0.163i)25-s + (−0.0505 + 0.544i)26-s + (−0.404 + 0.368i)29-s + (−0.273 − 0.961i)32-s + ⋯
L(s)  = 1  + (0.932 − 0.361i)2-s + (0.739 − 0.673i)4-s + (0.397 + 0.798i)5-s + (0.445 − 0.895i)8-s + (−0.273 + 0.961i)9-s + (0.658 + 0.600i)10-s + (−0.243 + 0.489i)13-s + (0.0922 − 0.995i)16-s + (0.0922 − 0.995i)17-s + (0.0922 + 0.995i)18-s + (0.831 + 0.322i)20-s + (0.123 − 0.163i)25-s + (−0.0505 + 0.544i)26-s + (−0.404 + 0.368i)29-s + (−0.273 − 0.961i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.876954343\)
\(L(\frac12)\) \(\approx\) \(1.876954343\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.932 + 0.361i)T \)
17 \( 1 + (-0.0922 + 0.995i)T \)
good3 \( 1 + (0.273 - 0.961i)T^{2} \)
5 \( 1 + (-0.397 - 0.798i)T + (-0.602 + 0.798i)T^{2} \)
7 \( 1 + (0.850 + 0.526i)T^{2} \)
11 \( 1 + (-0.0922 + 0.995i)T^{2} \)
13 \( 1 + (0.243 - 0.489i)T + (-0.602 - 0.798i)T^{2} \)
19 \( 1 + (-0.739 - 0.673i)T^{2} \)
23 \( 1 + (0.850 + 0.526i)T^{2} \)
29 \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \)
31 \( 1 + (0.602 + 0.798i)T^{2} \)
37 \( 1 + (1.83 - 0.342i)T + (0.932 - 0.361i)T^{2} \)
41 \( 1 + (0.537 + 0.711i)T + (-0.273 + 0.961i)T^{2} \)
43 \( 1 + (0.982 - 0.183i)T^{2} \)
47 \( 1 + (0.850 - 0.526i)T^{2} \)
53 \( 1 + (-0.465 + 1.63i)T + (-0.850 - 0.526i)T^{2} \)
59 \( 1 + (-0.445 + 0.895i)T^{2} \)
61 \( 1 + (-1.02 - 0.634i)T + (0.445 + 0.895i)T^{2} \)
67 \( 1 + (-0.739 - 0.673i)T^{2} \)
71 \( 1 + (0.850 + 0.526i)T^{2} \)
73 \( 1 + (0.0505 - 0.544i)T + (-0.982 - 0.183i)T^{2} \)
79 \( 1 + (-0.739 - 0.673i)T^{2} \)
83 \( 1 + (0.273 + 0.961i)T^{2} \)
89 \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \)
97 \( 1 + (0.404 - 1.42i)T + (-0.850 - 0.526i)T^{2} \)
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

−10.26503664583973994359755726551, −9.435498069429516014252589506344, −8.246417730016807730480404496638, −7.04857651971487567854503045679, −6.73304006883648306222873089043, −5.44901632063214513737841555527, −4.98137871149347705318385217063, −3.71529688182733058189612833105, −2.71547983211492471513436241733, −1.92048349028876229258633561738, 1.64028543105429784179796503169, 3.07375947480542724242307597521, 3.95057420323977737426872435840, 4.99059495217311847748369296939, 5.73323825440418030526315591180, 6.43107381543176030387176519927, 7.41049173208504637475188594068, 8.377545859125564949811489542264, 9.003847896923087837545160013800, 10.02576455001162777988856303726

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