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Properties

Label 2-1134-63.25-c1-0-29
Degree $2$
Conductor $1134$
Sign $0.149 + 0.988i$
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 + (1.62 − 2.09i)7-s + 8-s + (−2.12 − 3.67i)11-s + (−3.12 − 5.40i)13-s + (1.62 − 2.09i)14-s + 16-s + (−3.12 − 5.40i)19-s + (−2.12 − 3.67i)22-s + (−3.62 + 6.27i)23-s + (2.5 + 4.33i)25-s + (−3.12 − 5.40i)26-s + (1.62 − 2.09i)28-s + (2.12 − 3.67i)29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.612 − 0.790i)7-s + 0.353·8-s + (−0.639 − 1.10i)11-s + (−0.865 − 1.49i)13-s + (0.433 − 0.558i)14-s + 0.250·16-s + (−0.716 − 1.24i)19-s + (−0.452 − 0.783i)22-s + (−0.755 + 1.30i)23-s + (0.5 + 0.866i)25-s + (−0.612 − 1.06i)26-s + (0.306 − 0.395i)28-s + (0.393 − 0.682i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.149 + 0.988i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.149 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.268129367\)
\(L(\frac12)\) \(\approx\) \(2.268129367\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.09i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.12 + 5.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.12 + 5.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.62 - 6.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.12 + 3.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.757T + 31T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.74 - 4.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 + 5.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + (2.12 - 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6.24T + 61T^{2} \)
67 \( 1 + 8.24T + 67T^{2} \)
71 \( 1 + 7.24T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.24T + 79T^{2} \)
83 \( 1 + (3.87 - 6.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.74 - 4.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.24 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.858842942341349667702166568357, −8.635019593224541922075595431679, −7.74817982595981113527426077899, −7.29959615926149841982441058293, −6.01735242760380177635142130855, −5.30651409957930954727202738140, −4.49402575824741178732324321700, −3.37404656139230669622979924582, −2.48748807354091873390391181193, −0.76163926501138266099163050222, 1.99917534018284401958460990679, 2.48119321279969193552510094875, 4.27239486488439598139048340080, 4.60616053109621776014865472143, 5.69341668886969722400699768909, 6.56176524892409248650293616702, 7.42563484188372690166690466260, 8.290474993462838949343679754638, 9.164548220864788958455728569199, 10.16395039975470678718216560020

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