Properties

Label 2-1110-37.11-c1-0-11
Degree $2$
Conductor $1110$
Sign $0.671 + 0.740i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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 + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s − 0.999i·6-s + (−2.37 + 4.12i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 5.36·11-s + (0.499 + 0.866i)12-s + (−1.67 − 0.968i)13-s − 4.75i·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (3.25 − 1.87i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s − 0.408i·6-s + (−0.899 + 1.55i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 1.61·11-s + (0.144 + 0.249i)12-s + (−0.465 − 0.268i)13-s − 1.27i·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.789 − 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3071558488\)
\(L(\frac12)\) \(\approx\) \(0.3071558488\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-2.60 - 5.49i)T \)
good7 \( 1 + (2.37 - 4.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.36T + 11T^{2} \)
13 \( 1 + (1.67 + 0.968i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.25 + 1.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 0.945i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.204iT - 23T^{2} \)
29 \( 1 + 2.07iT - 29T^{2} \)
31 \( 1 + 1.95iT - 31T^{2} \)
41 \( 1 + (-3.63 + 6.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 1.74iT - 43T^{2} \)
47 \( 1 - 8.35T + 47T^{2} \)
53 \( 1 + (6.35 + 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.23 - 4.75i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.60 + 2.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.26 - 3.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.89 + 8.47i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 + (10.1 + 5.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.10 + 3.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.32 + 3.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.30iT - 97T^{2} \)
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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

−9.698045715729860655188915451566, −9.017030848971301357470038611396, −8.104977337888073380152857350805, −7.47630180897819332556131729340, −6.19021983968369704985879771970, −5.52124490521754299047031018879, −4.91135253100472608198785927103, −3.23665260337974752985247418877, −2.44205441632028104908802624983, −0.21349065529766710323582832363, 0.961064926750539752848348171426, 2.62313325565697679670792683724, 3.51362307114932215959367520624, 4.64295055116221871851464865799, 5.91326745457609025289615361275, 6.95971300876286078041460766302, 7.55618361408044278284836860974, 7.972562259833921603034842483641, 9.331177789487980024429068482409, 10.14448663368817007001663158162

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