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Properties

Label 2-1134-63.47-c1-0-27
Degree $2$
Conductor $1134$
Sign $-0.592 + 0.805i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.5 − 0.866i)7-s − 0.999i·8-s − 3i·11-s + (3 − 1.73i)13-s + (−2.59 + 0.500i)14-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)22-s − 5·25-s + (1.73 − 3i)26-s + (−1.99 + 1.73i)28-s + (−7.79 − 4.5i)29-s + (1.5 + 0.866i)31-s + (−0.866 − 0.499i)32-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.944 − 0.327i)7-s − 0.353i·8-s − 0.904i·11-s + (0.832 − 0.480i)13-s + (−0.694 + 0.133i)14-s + (−0.125 − 0.216i)16-s + (−0.319 − 0.553i)22-s − 25-s + (0.339 − 0.588i)26-s + (−0.377 + 0.327i)28-s + (−1.44 − 0.835i)29-s + (0.269 + 0.155i)31-s + (−0.153 − 0.0883i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.675323556\)
\(L(\frac12)\) \(\approx\) \(1.675323556\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
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 \( 1 - 23T^{2} \)
29 \( 1 + (7.79 + 4.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
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 \( 1 + (5.19 + 9i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12 + 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
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

−9.708356230947816217691477147035, −8.805524813968759433522375740248, −7.84829407852601714919582293497, −6.82323949825183464593154193507, −5.97881594613866622739553479565, −5.40497848118695277787030388804, −3.84264637602976889375794164533, −3.54824882580925946953713407112, −2.22782047255857120914711178020, −0.57799652806191971808712768089, 1.83854381754686040360242480343, 3.12435435230595439166457848764, 3.98099765188957114360851619991, 4.97598731954459548205680918099, 6.01649664818177420919140427332, 6.59179256408968710387520566621, 7.45815590869819033836293590383, 8.386708974266617935854514909436, 9.368927000242760297256368926129, 9.918611074947572669823470137215

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