↑

Properties

Label 2-792-11.9-c1-0-13
Degree $2$
Conductor $792$
Sign $-0.998 + 0.0540i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
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 − 3.23i)5-s + (−0.703 − 0.510i)7-s + (−3.16 − 0.998i)11-s + (0.866 − 2.66i)13-s + (2.25 + 6.93i)17-s + (−1.92 + 1.40i)19-s − 7.44·23-s + (−5.34 + 3.88i)25-s + (−5.93 − 4.31i)29-s + (−0.816 + 2.51i)31-s + (−0.914 + 2.81i)35-s + (−0.834 − 0.605i)37-s + (2.34 − 1.70i)41-s + 2.18·43-s + (−3.19 + 2.31i)47-s + ⋯
L(s)  = 1  + (−0.470 − 1.44i)5-s + (−0.265 − 0.193i)7-s + (−0.953 − 0.301i)11-s + (0.240 − 0.739i)13-s + (0.546 + 1.68i)17-s + (−0.442 + 0.321i)19-s − 1.55·23-s + (−1.06 + 0.776i)25-s + (−1.10 − 0.801i)29-s + (−0.146 + 0.451i)31-s + (−0.154 + 0.475i)35-s + (−0.137 − 0.0996i)37-s + (0.365 − 0.265i)41-s + 0.333·43-s + (−0.465 + 0.338i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.998 + 0.0540i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.998 + 0.0540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0137548 - 0.508493i\)
\(L(\frac12)\) \(\approx\) \(0.0137548 - 0.508493i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (3.16 + 0.998i)T \)
good5 \( 1 + (1.05 + 3.23i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.703 + 0.510i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.866 + 2.66i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.25 - 6.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.92 - 1.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.44T + 23T^{2} \)
29 \( 1 + (5.93 + 4.31i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.816 - 2.51i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.834 + 0.605i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.34 + 1.70i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.18T + 43T^{2} \)
47 \( 1 + (3.19 - 2.31i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.19 + 6.76i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.54 - 4.75i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.832 + 2.56i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + (-0.596 - 1.83i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.03 + 2.93i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.15 + 12.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.0601 - 0.185i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.18T + 89T^{2} \)
97 \( 1 + (-1.54 + 4.74i)T + (-78.4 - 57.0i)T^{2} \)
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

−9.992523948752305038816289419361, −8.784331246963031317258875309273, −8.158874862472973242037122841804, −7.67496929403051075164763646838, −6.03182106436483903391961855132, −5.49956188151461459044100325038, −4.30763373822758850330302241768, −3.53548500413385691156175991579, −1.78951835528885506111808824122, −0.24118586927051563360971080410, 2.29372648021937097200733043607, 3.13016710645240185708633184290, 4.21940943522589201403484630174, 5.47910652794093220208913384587, 6.49456839048564368049388303584, 7.30660267415812276049613090515, 7.83739660117399585697164107296, 9.150888610911907698887840174147, 9.938856178707625906927653122925, 10.72593334246056543349864218649

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