Properties

Label 2-1134-63.25-c1-0-12
Degree $2$
Conductor $1134$
Sign $0.888 - 0.458i$
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 + (2.5 + 0.866i)7-s − 8-s + (2 + 3.46i)13-s + (−2.5 − 0.866i)14-s + 16-s + (3 − 5.19i)17-s + (−1 − 1.73i)19-s + (−1.5 + 2.59i)23-s + (2.5 + 4.33i)25-s + (−2 − 3.46i)26-s + (2.5 + 0.866i)28-s + (−3 + 5.19i)29-s + 5·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.944 + 0.327i)7-s − 0.353·8-s + (0.554 + 0.960i)13-s + (−0.668 − 0.231i)14-s + 0.250·16-s + (0.727 − 1.26i)17-s + (−0.229 − 0.397i)19-s + (−0.312 + 0.541i)23-s + (0.5 + 0.866i)25-s + (−0.392 − 0.679i)26-s + (0.472 + 0.163i)28-s + (−0.557 + 0.964i)29-s + 0.898·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.888 - 0.458i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.344404672\)
\(L(\frac12)\) \(\approx\) \(1.344404672\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
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 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.646826547508947126106558904146, −9.084912266370969053508678841913, −8.349843226609282130415069876291, −7.45400219865103127652679304904, −6.80832067136748689429010235194, −5.61715880863236205190427037923, −4.84657227239813815111995490214, −3.56212991025573369338545002227, −2.28677352455988310763422505300, −1.18571287118490488614326846276, 0.919365518272345084312670568734, 2.09709791278411172227671144391, 3.46058886128328040526502467195, 4.51186295546731558450092998557, 5.69775137207133593565392116214, 6.40811020488025342340455604687, 7.60498116260930810576729796684, 8.218229652425791815592869543154, 8.629019942972356023061823142025, 10.00762452544365622256816864013

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