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Properties

Label 2-1134-21.5-c1-0-25
Degree $2$
Conductor $1134$
Sign $0.993 + 0.112i$
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 + (1.75 − 3.03i)5-s + (2.30 + 1.29i)7-s + 0.999i·8-s + (3.03 − 1.75i)10-s + (4.72 − 2.72i)11-s + 2.51i·13-s + (1.35 + 2.27i)14-s + (−0.5 + 0.866i)16-s + (−0.852 − 1.47i)17-s + (−4.19 − 2.42i)19-s + 3.50·20-s + 5.45·22-s + (−4.63 − 2.67i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.784 − 1.35i)5-s + (0.872 + 0.487i)7-s + 0.353i·8-s + (0.960 − 0.554i)10-s + (1.42 − 0.822i)11-s + 0.698i·13-s + (0.362 + 0.607i)14-s + (−0.125 + 0.216i)16-s + (−0.206 − 0.358i)17-s + (−0.963 − 0.556i)19-s + 0.784·20-s + 1.16·22-s + (−0.966 − 0.558i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.993 + 0.112i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.021595771\)
\(L(\frac12)\) \(\approx\) \(3.021595771\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 - 1.29i)T \)
good5 \( 1 + (-1.75 + 3.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.72 + 2.72i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.51iT - 13T^{2} \)
17 \( 1 + (0.852 + 1.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.19 + 2.42i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.63 + 2.67i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.12iT - 29T^{2} \)
31 \( 1 + (5.25 - 3.03i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.14 + 7.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.75T + 41T^{2} \)
43 \( 1 - 6.45T + 43T^{2} \)
47 \( 1 + (4.31 - 7.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.0 + 5.81i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.94 - 6.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.26 + 4.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.895 - 1.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.791iT - 71T^{2} \)
73 \( 1 + (11.5 - 6.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.40 - 4.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.465T + 83T^{2} \)
89 \( 1 + (-1.09 + 1.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−9.372977444441237557215275496801, −8.872319197795821964236277915852, −8.463594602799650775499211663902, −7.16757407821414584619906232198, −6.15323145609560227072836295728, −5.58198684276465735778608221928, −4.61796404976065960886190041730, −4.03557449317919696822022223565, −2.30894115404906004570170759853, −1.29978140983617055589798334494, 1.63683661410655732533154616470, 2.38939879587358630529344962115, 3.77186677220889835043558132773, 4.35013413551917151659158026827, 5.77133859874161060944622339545, 6.30469141656934179246597470513, 7.18099411202412157056722229589, 7.988529286626936517385046111358, 9.319643619188097247345846175638, 10.14174698950144575426028084185

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