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Properties

Label 2-1134-63.25-c1-0-7
Degree $2$
Conductor $1134$
Sign $0.0477 - 0.998i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−1.5 + 2.59i)5-s + (−0.5 − 2.59i)7-s + 8-s + (−1.5 + 2.59i)10-s + (2 + 3.46i)13-s + (−0.5 − 2.59i)14-s + 16-s + (−3 + 5.19i)17-s + (2 + 3.46i)19-s + (−1.5 + 2.59i)20-s + (−3 + 5.19i)23-s + (−2 − 3.46i)25-s + (2 + 3.46i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.670 + 1.16i)5-s + (−0.188 − 0.981i)7-s + 0.353·8-s + (−0.474 + 0.821i)10-s + (0.554 + 0.960i)13-s + (−0.133 − 0.694i)14-s + 0.250·16-s + (−0.727 + 1.26i)17-s + (0.458 + 0.794i)19-s + (−0.335 + 0.580i)20-s + (−0.625 + 1.08i)23-s + (−0.400 − 0.692i)25-s + (0.392 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0477 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.0477 - 0.998i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.0477 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934749980\)
\(L(\frac12)\) \(\approx\) \(1.934749980\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 17T + 79T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.38886397933334988193916629202, −9.323973478791458326878873567121, −7.959001138170310168447772434151, −7.50967069718777235638547235447, −6.45375723586784867457889984428, −6.15101073263977076467663350953, −4.47542373088997820439236477953, −3.87182470730173416153711415545, −3.14216588243346435465642155812, −1.67467004161053159171230270879, 0.67872419458169656971759809767, 2.42667620499939067781944617911, 3.38537071491918194082525401274, 4.69758357275940886803603557316, 5.03705731211878736077582173957, 6.08746777451634720510656709676, 6.99933598987310551273753361842, 8.184663599171142558076462388587, 8.623408731786003754530177064383, 9.475663151672731156418072590876

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