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Properties

Label 2-1134-63.38-c1-0-11
Degree $2$
Conductor $1134$
Sign $0.942 - 0.333i$
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 + (0.358 − 0.621i)5-s + (−1.62 + 2.09i)7-s + i·8-s + (−0.621 − 0.358i)10-s + (−2.59 + 1.5i)11-s + (2.12 − 1.22i)13-s + (2.09 + 1.62i)14-s + 16-s + (2.95 − 5.12i)17-s + (−5.12 + 2.95i)19-s + (−0.358 + 0.621i)20-s + (1.5 + 2.59i)22-s + (3.67 + 2.12i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.160 − 0.277i)5-s + (−0.612 + 0.790i)7-s + 0.353i·8-s + (−0.196 − 0.113i)10-s + (−0.783 + 0.452i)11-s + (0.588 − 0.339i)13-s + (0.558 + 0.433i)14-s + 0.250·16-s + (0.717 − 1.24i)17-s + (−1.17 + 0.678i)19-s + (−0.0802 + 0.138i)20-s + (0.319 + 0.553i)22-s + (0.766 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.226798634\)
\(L(\frac12)\) \(\approx\) \(1.226798634\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (-0.358 + 0.621i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.12 - 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.27 - 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.08iT - 31T^{2} \)
37 \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.91 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.121 + 0.210i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 + (-6.27 - 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.06T + 59T^{2} \)
61 \( 1 - 1.01iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (1.24 + 0.717i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 + (-3.31 + 5.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.7 - 6.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.991784403542382656574144866572, −9.058204990033000420363382623263, −8.544187614125926225830162065874, −7.47919980585530422290411014309, −6.40645814520685630122603673888, −5.37961813961550618918555055797, −4.78356210183304397363760197505, −3.31096574893145471880048906224, −2.68979758401377467839563811222, −1.26556588765282041492062990698, 0.60063687409070419335817745575, 2.52632723036624722556386258363, 3.76425553126840741297476579947, 4.52788484681936984507917023106, 5.84827645329611366531586849229, 6.38401348296098954302209004879, 7.17388745266210941972436011931, 8.165599648613906040534261179844, 8.704816001510320004894021401053, 9.833106861616983874732536195631

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