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Properties

Label 2-1134-63.58-c1-0-8
Degree $2$
Conductor $1134$
Sign $-0.0477 - 0.998i$
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 + (1.5 + 2.59i)5-s + (−0.5 + 2.59i)7-s − 8-s + (−1.5 − 2.59i)10-s + (2 − 3.46i)13-s + (0.5 − 2.59i)14-s + 16-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s + (1.5 + 2.59i)20-s + (3 + 5.19i)23-s + (−2 + 3.46i)25-s + (−2 + 3.46i)26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.670 + 1.16i)5-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (−0.474 − 0.821i)10-s + (0.554 − 0.960i)13-s + (0.133 − 0.694i)14-s + 0.250·16-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s + (0.335 + 0.580i)20-s + (0.625 + 1.08i)23-s + (−0.400 + 0.692i)25-s + (−0.392 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.0477 - 0.998i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.0477 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.295622440\)
\(L(\frac12)\) \(\approx\) \(1.295622440\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
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 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 17T + 79T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.14956331170890471153454533301, −9.280973068077343621633766763669, −8.424082234198822255006942348559, −7.67224223603255596966376839082, −6.58581498595970786976203595444, −6.05661535867833975740730199557, −5.22965819784247440777917897026, −3.32945796171510404594180785747, −2.79093233604965698282726990498, −1.51170455996311713644398502063, 0.77593272133235155404951997776, 1.69472685761918745251955579675, 3.24282323564344471485772370994, 4.48179559245567463677681177920, 5.31972660538423526136429841017, 6.41991919044478154507317534546, 7.16444692084898744562597660190, 8.124997421604800971702105607134, 8.909845541773668939683782046187, 9.563879823023061226078177454463

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