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Properties

Label 2-1134-9.4-c1-0-1
Degree $2$
Conductor $1134$
Sign $-0.939 - 0.342i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2 + 3.46i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s + 3.99·10-s + (2 + 3.46i)11-s + (−1.5 + 2.59i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 7·17-s + 2·19-s + (−1.99 − 3.46i)20-s + (1.99 − 3.46i)22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.894 + 1.54i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s + 1.26·10-s + (0.603 + 1.04i)11-s + (−0.416 + 0.720i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 1.69·17-s + 0.458·19-s + (−0.447 − 0.774i)20-s + (0.426 − 0.738i)22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4860952950\)
\(L(\frac12)\) \(\approx\) \(0.4860952950\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-4 - 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.19800382182761284500871905154, −9.537154856141914626924019034323, −8.623395974887380269057849262056, −7.62560578139108787484362907704, −6.98737956701071295475766882623, −6.34275230819685207099896925689, −4.57813493017129180577693320497, −4.01643135865326696771623267185, −2.79161018573645939939805550299, −2.03530330506933544544284006759, 0.25340362835138770231659021176, 1.34615706998305570310593181944, 3.37378943935730196956077133266, 4.50873365552984938092675615328, 5.01379203689473519987847193449, 6.09717354140860251297452775739, 7.12475123692516587405018486480, 7.971331166667962069064275378578, 8.686338940163168085993602952564, 8.983328446436336103223658547695

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