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Properties

Label 2-1134-9.7-c1-0-21
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: $\chi_{1134} (379, \cdot )$
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

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

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