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Properties

Label 2-1110-185.159-c1-0-25
Degree $2$
Conductor $1110$
Sign $0.0878 + 0.996i$
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  + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−1.56 + 1.59i)5-s − 0.999i·6-s + (−0.998 − 0.576i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−2.16 − 0.553i)10-s + 1.60·11-s + (0.866 − 0.499i)12-s + (−1.89 + 3.28i)13-s − 1.15i·14-s + (2.15 − 0.603i)15-s + (−0.5 − 0.866i)16-s + (−3.28 − 5.69i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.698 + 0.715i)5-s − 0.408i·6-s + (−0.377 − 0.217i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.685 − 0.175i)10-s + 0.482·11-s + (0.249 − 0.144i)12-s + (−0.526 + 0.911i)13-s − 0.308i·14-s + (0.555 − 0.155i)15-s + (−0.125 − 0.216i)16-s + (−0.797 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.0878 + 0.996i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.0878 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3751662796\)
\(L(\frac12)\) \(\approx\) \(0.3751662796\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (1.56 - 1.59i)T \)
37 \( 1 + (6.07 - 0.267i)T \)
good7 \( 1 + (0.998 + 0.576i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.60T + 11T^{2} \)
13 \( 1 + (1.89 - 3.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.28 + 5.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.174 + 0.100i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.633T + 23T^{2} \)
29 \( 1 - 5.73iT - 29T^{2} \)
31 \( 1 + 8.34iT - 31T^{2} \)
41 \( 1 + (-4.00 + 6.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.95T + 43T^{2} \)
47 \( 1 + 4.65iT - 47T^{2} \)
53 \( 1 + (-7.47 + 4.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.3 + 5.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 + 6.54i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.33 + 5.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.52 - 9.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.32iT - 73T^{2} \)
79 \( 1 + (-6.91 - 3.99i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.4 - 7.21i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-15.6 + 9.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.93T + 97T^{2} \)
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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

−9.556061502641009567573550815678, −8.760139719620071694286598675363, −7.61495995468182641671797185774, −6.83500083057529703187410123424, −6.73068872621993696906494077171, −5.40283034269369920910524205118, −4.45876349396332873652486239085, −3.61005752239806984410130333558, −2.34723461027130848958449279108, −0.16376100877268086375514783220, 1.35010183261987820898663277644, 2.95822623308380052902020016806, 4.00185579553885432330131946281, 4.66768875546494421465946957290, 5.63488704290274233721793964851, 6.43460231521877516713982666172, 7.62634592472881818419707540179, 8.626807567993831346610480832836, 9.241076349258914141656733626287, 10.29562305606379227763842872228

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