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Properties

Label 2-1134-63.5-c1-0-2
Degree $2$
Conductor $1134$
Sign $-0.690 - 0.723i$
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  + i·2-s − 4-s + (−0.866 − 1.5i)5-s + (−2.5 − 0.866i)7-s − i·8-s + (1.5 − 0.866i)10-s + (0.866 − 2.5i)14-s + 16-s + (1.73 + 3i)17-s + (6 + 3.46i)19-s + (0.866 + 1.5i)20-s + (−5.19 + 3i)23-s + (1 − 1.73i)25-s + (2.5 + 0.866i)28-s + (−7.79 + 4.5i)29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.387 − 0.670i)5-s + (−0.944 − 0.327i)7-s − 0.353i·8-s + (0.474 − 0.273i)10-s + (0.231 − 0.668i)14-s + 0.250·16-s + (0.420 + 0.727i)17-s + (1.37 + 0.794i)19-s + (0.193 + 0.335i)20-s + (−1.08 + 0.625i)23-s + (0.200 − 0.346i)25-s + (0.472 + 0.163i)28-s + (−1.44 + 0.835i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.690 - 0.723i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.690 - 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7490609738\)
\(L(\frac12)\) \(\approx\) \(0.7490609738\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.73 - 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.79 - 4.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.73 - 3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + (-2.59 + 1.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + (-8.66 - 15i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 - 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.798071355947411066889536143805, −9.374744089638480587830317412753, −8.259136420453439668681487436407, −7.73901842230059655192988066793, −6.80996001795398050724283337650, −5.90792420448656033212758635368, −5.16049894023288694875246249067, −3.98199708061392832995914317775, −3.31095347315847509005594815213, −1.30265569086316676921000972987, 0.35455714758699262879196960302, 2.25780566444298190212407930663, 3.17990500058952823133106132510, 3.87981853174853643816444169507, 5.19604708337470404041773207701, 6.06801380903661751535780622912, 7.15984683564676494845300859444, 7.75531563332760740785672295061, 9.082302454573372195502800416962, 9.515443721669730561303310928527

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