↑

Properties

Label 2-935-935.274-c0-0-2
Degree $2$
Conductor $935$
Sign $0.856 - 0.516i$
Analytic cond. $0.466625$
Root an. cond. $0.683100$
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  + (0.785 + 0.785i)2-s + 0.234i·4-s + (−0.382 − 0.923i)5-s + (−0.636 + 1.53i)7-s + (0.601 − 0.601i)8-s + (0.707 − 0.707i)9-s + (0.425 − 1.02i)10-s + (0.923 + 0.382i)11-s + 0.390i·13-s + (−1.70 + 0.707i)14-s + 1.17·16-s + (0.195 − 0.980i)17-s + 1.11·18-s + (0.216 − 0.0897i)20-s + (0.425 + 1.02i)22-s + ⋯
L(s)  = 1  + (0.785 + 0.785i)2-s + 0.234i·4-s + (−0.382 − 0.923i)5-s + (−0.636 + 1.53i)7-s + (0.601 − 0.601i)8-s + (0.707 − 0.707i)9-s + (0.425 − 1.02i)10-s + (0.923 + 0.382i)11-s + 0.390i·13-s + (−1.70 + 0.707i)14-s + 1.17·16-s + (0.195 − 0.980i)17-s + 1.11·18-s + (0.216 − 0.0897i)20-s + (0.425 + 1.02i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(0.466625\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :0),\ 0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479066212\)
\(L(\frac12)\) \(\approx\) \(1.479066212\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-0.195 + 0.980i)T \)
good2 \( 1 + (-0.785 - 0.785i)T + iT^{2} \)
3 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 - 0.390iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.785 - 0.785i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.750 + 1.81i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
89 \( 1 - 1.84iT - T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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

−9.999126623805221614245356018770, −9.264825125909709183674495032521, −8.894306196289080095902451604458, −7.52429233496388678766010980881, −6.73694887053241838192015955285, −5.98467623609695409546714856499, −5.14414192898545215367569120701, −4.35140850844353869435920537684, −3.36561947646640124644630647906, −1.55727103053502811059764729118, 1.66073695437158975569520679988, 3.12685343093904596544341993726, 3.86424254377010002409429429062, 4.30943850606878269506385122604, 5.79419437049137634272842337268, 6.95994289248921091847837286943, 7.43430381662100464719422515638, 8.349095142530015195593663732675, 9.889465781245711514117062959481, 10.45621746300991561094621675546

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