Properties

Label 2-1134-63.5-c1-0-24
Degree $2$
Conductor $1134$
Sign $-0.971 + 0.235i$
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  i·2-s − 4-s + (2 − 1.73i)7-s + i·8-s + (−2.59 − 1.5i)11-s + (−3 − 1.73i)13-s + (−1.73 − 2i)14-s + 16-s + (−1.5 + 2.59i)22-s + (2.5 − 4.33i)25-s + (−1.73 + 3i)26-s + (−2 + 1.73i)28-s + (−7.79 + 4.5i)29-s − 1.73i·31-s i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.755 − 0.654i)7-s + 0.353i·8-s + (−0.783 − 0.452i)11-s + (−0.832 − 0.480i)13-s + (−0.462 − 0.534i)14-s + 0.250·16-s + (−0.319 + 0.553i)22-s + (0.5 − 0.866i)25-s + (−0.339 + 0.588i)26-s + (−0.377 + 0.327i)28-s + (−1.44 + 0.835i)29-s − 0.311i·31-s − 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.971 + 0.235i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.971 + 0.235i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.009882333\)
\(L(\frac12)\) \(\approx\) \(1.009882333\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.79 - 4.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.19 + 9i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.19T + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + (2.59 + 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.5 - 4.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.575541932364415972448828128291, −8.676334171731579765145068400518, −7.80370115128048126707051384336, −7.23576182202318769637400483350, −5.77776426467531145764204439191, −5.00521793472112433560827518729, −4.11869238455494178236587778425, −3.01050181257853874251539141586, −1.93330307150520021880281721753, −0.42873801839899749518344730018, 1.77018177836370411429208267102, 2.99045513128380565268817874521, 4.52752512045613882204835694933, 5.03863332572544049951876785482, 5.95047781466844273960731481680, 6.95519500049058295497228695704, 7.80343471656103348883854036823, 8.297813433347926796862285181900, 9.453365845514240146073432787125, 9.810806799168209878810693863178

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