Properties

Label 2-1183-91.32-c0-0-1
Degree $2$
Conductor $1183$
Sign $0.222 + 0.974i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
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  + (0.707 − 0.707i)2-s + (−0.5 − 0.866i)3-s + (0.965 + 0.258i)5-s + (−0.965 − 0.258i)6-s + (0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.258 + 0.965i)11-s + (−0.500 − 0.866i)14-s + (−0.258 − 0.965i)15-s + 1.00·16-s + i·17-s + (−0.258 − 0.965i)19-s + (−0.965 + 0.258i)21-s + (0.500 + 0.866i)22-s + i·23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.5 − 0.866i)3-s + (0.965 + 0.258i)5-s + (−0.965 − 0.258i)6-s + (0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.258 + 0.965i)11-s + (−0.500 − 0.866i)14-s + (−0.258 − 0.965i)15-s + 1.00·16-s + i·17-s + (−0.258 − 0.965i)19-s + (−0.965 + 0.258i)21-s + (0.500 + 0.866i)22-s + i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.222 + 0.974i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.222 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.575824092\)
\(L(\frac12)\) \(\approx\) \(1.575824092\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
97 \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \)
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

−10.13544798173690550829078335673, −9.150768607010025768276515032300, −7.76586847540454900724549617511, −7.30927089471544553255988697350, −6.40717620226909456341458865201, −5.51355345027283307116005215748, −4.50243432948906182873191942930, −3.63940421250075811332234338697, −2.22934974392979823974876497369, −1.56319410062699380747270530916, 1.76946707379895979292838663226, 3.24055530891732823347953491505, 4.64761400146674548021442464693, 5.07865768492404828620873706143, 5.87335815210409334268220758046, 6.22349584440960533782044556852, 7.53926784363859690239604208978, 8.615328908406025913453404948398, 9.399983469892542045444680505888, 10.19824238223610739707654278657

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