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Properties

Label 2-1134-21.5-c1-0-23
Degree $2$
Conductor $1134$
Sign $0.779 - 0.626i$
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 + (−0.894 + 1.54i)5-s + (2.54 − 0.731i)7-s + 0.999i·8-s + (−1.54 + 0.894i)10-s + (4.52 − 2.61i)11-s − 5.03i·13-s + (2.56 + 0.637i)14-s + (−0.5 + 0.866i)16-s + (1.13 + 1.96i)17-s + (−3.45 − 1.99i)19-s − 1.78·20-s + 5.22·22-s + (6.84 + 3.94i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.400 + 0.692i)5-s + (0.960 − 0.276i)7-s + 0.353i·8-s + (−0.489 + 0.282i)10-s + (1.36 − 0.788i)11-s − 1.39i·13-s + (0.686 + 0.170i)14-s + (−0.125 + 0.216i)16-s + (0.274 + 0.475i)17-s + (−0.792 − 0.457i)19-s − 0.400·20-s + 1.11·22-s + (1.42 + 0.823i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.779 - 0.626i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.779 - 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.615509198\)
\(L(\frac12)\) \(\approx\) \(2.615509198\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 + 0.731i)T \)
good5 \( 1 + (0.894 - 1.54i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.52 + 2.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.03iT - 13T^{2} \)
17 \( 1 + (-1.13 - 1.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 + 1.99i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.84 - 3.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.38iT - 29T^{2} \)
31 \( 1 + (-6.73 + 3.88i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.74 - 6.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.03T + 41T^{2} \)
43 \( 1 + 8.81T + 43T^{2} \)
47 \( 1 + (-2.07 + 3.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.93 - 1.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.43 - 9.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.65 - 2.11i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.705 + 1.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.11iT - 71T^{2} \)
73 \( 1 + (4.06 - 2.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.53 + 6.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 + (2.97 - 5.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 19.3iT - 97T^{2} \)
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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

−10.12762305869095480151313829908, −8.688065036088395281566844652573, −8.281074563358022895135236804466, −7.22351848592413123982875940572, −6.65786658675370027102060846413, −5.61586811486415926107186465803, −4.77165380407201736094569654601, −3.65556131965109286767656176635, −3.03224293747567702048846042760, −1.28967696199368096669167169258, 1.26914581452596244062827852409, 2.19829067276845526748043796031, 3.78594680300589778044281151388, 4.61127555281545280914391545873, 4.97854169351477613609240411315, 6.47401117249279376022950649681, 6.96869763015342659614617086106, 8.302589419171062804157948502480, 8.878565180805811532690278860143, 9.681031337789832701136485642589

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