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Properties

Label 2-1134-63.25-c1-0-17
Degree $2$
Conductor $1134$
Sign $0.995 - 0.0954i$
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 + (1.5 + 2.59i)11-s + (2 + 3.46i)13-s + (−0.5 + 2.59i)14-s + 16-s + (2 + 3.46i)19-s + (1.5 − 2.59i)20-s + (1.5 + 2.59i)22-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 + (0.452 + 0.783i)11-s + (0.554 + 0.960i)13-s + (−0.133 + 0.694i)14-s + 0.250·16-s + (0.458 + 0.794i)19-s + (0.335 − 0.580i)20-s + (0.319 + 0.553i)22-s + (−0.400 − 0.692i)25-s + (0.392 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.995 - 0.0954i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.995 - 0.0954i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.951353218\)
\(L(\frac12)\) \(\approx\) \(2.951353218\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.570657857977124175063127260813, −9.161025741875620856237567189831, −8.333112414378675571933301712675, −7.18058646339383269467834561703, −6.11254944491573633372726538693, −5.62828129582997066827082510487, −4.66991497281474670468314912313, −3.87604403996193729340221152206, −2.35709993721071187179039999045, −1.50299735085192217433994096264, 1.23364894125359973306798330016, 2.97445783527431546374993377278, 3.28996170527260319280182847733, 4.57185517025400394800433738777, 5.69154292497998232116297210208, 6.45462975451397560921282141585, 7.00989301982595332422830086580, 7.936817699597560586236717736410, 9.076399007620714247514914026855, 10.13228519038860546160586425906

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