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Properties

Label 2-1110-37.11-c1-0-23
Degree $2$
Conductor $1110$
Sign $-0.529 + 0.848i$
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 + (1.04 − 1.81i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + 5.41·11-s + (−0.499 − 0.866i)12-s + (−4.29 − 2.48i)13-s − 2.09i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (−3.13 + 1.81i)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 + (0.395 − 0.684i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s + 1.63·11-s + (−0.144 − 0.249i)12-s + (−1.19 − 0.688i)13-s − 0.559i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.761 + 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.529 + 0.848i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.529 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.398017465\)
\(L(\frac12)\) \(\approx\) \(2.398017465\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.57 - 4.00i)T \)
good7 \( 1 + (-1.04 + 1.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 5.41T + 11T^{2} \)
13 \( 1 + (4.29 + 2.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.13 - 1.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.82 - 2.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.06iT - 23T^{2} \)
29 \( 1 - 1.37iT - 29T^{2} \)
31 \( 1 + 4.90iT - 31T^{2} \)
41 \( 1 + (3.54 - 6.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4.69iT - 43T^{2} \)
47 \( 1 - 1.42T + 47T^{2} \)
53 \( 1 + (1.07 + 1.86i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.3 + 7.11i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.87 + 1.65i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.09 - 1.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.21 + 12.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + (0.685 + 0.395i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.95 - 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.917 - 0.529i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.37iT - 97T^{2} \)
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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

−9.650965790006710675896185221738, −8.692341864157505878747788932423, −7.83057730505297754032284512668, −6.98701074497687193046954390375, −6.30819112746566203702914995721, −5.02466177972491082136322147428, −4.22160304717263659229241208422, −3.37345583195442020994500054793, −2.06741934976005711943841313198, −0.863159232560099236667152065827, 1.94199892540223044821469569206, 3.13590589416084505567126882204, 4.08225911641770959992611768063, 4.86953256723115035231872794919, 5.71969203050703993691316980739, 7.00191927872411084657677462198, 7.29028780344611766894777062305, 8.682123688329306078606287749985, 9.120507671005027638523719690425, 9.941507443764656228435389390554

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