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Properties

Label 2-1134-63.25-c1-0-14
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

\(L(1)\) \(\approx\) \(2.836936757\)
\(L(\frac12)\) \(\approx\) \(2.836936757\)
\(L(\frac{3}{2})\) 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

−10.02970648744646252098889400772, −8.747743310230295735259609702422, −8.433095102713595573631883100319, −7.20044604348840720991837768708, −6.44499856994458284274529736581, −5.57783172556145096946663709445, −4.56813176005885191995840177043, −3.96919643955467134984066752511, −2.54062012141973695952783034456, −1.55162904800395650213867468481, 1.14538182954629501756742180596, 2.55998803375001182095488532862, 3.57734765278609824741732005798, 4.75921206485804303885288800752, 5.18562359408352725886083039487, 6.39911864534848620749143303972, 7.13239069102917032161630753226, 8.129033338426144666038968600108, 8.689223489429598187327129036659, 10.03498125126303768559099391071

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