Properties

Label 2-1110-37.11-c1-0-25
Degree $2$
Conductor $1110$
Sign $-0.146 + 0.989i$
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 + (2.23 − 3.86i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 2·11-s + (−0.499 − 0.866i)12-s + (−0.866 − 0.5i)13-s − 4.46i·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−1.26 + 0.732i)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 + (0.843 − 1.46i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 0.603·11-s + (−0.144 − 0.249i)12-s + (−0.240 − 0.138i)13-s − 1.19i·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.307 + 0.177i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.146 + 0.989i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.923000336\)
\(L(\frac12)\) \(\approx\) \(2.923000336\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 5.5i)T \)
good7 \( 1 + (-2.23 + 3.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.26 - 0.732i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.59 + 1.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.73iT - 23T^{2} \)
29 \( 1 - 3.73iT - 29T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
41 \( 1 + (-2.09 + 3.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.19iT - 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 + (0.366 + 0.633i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.63 - 2.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.02 + 4.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 + 3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.401 + 0.696i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.80T + 73T^{2} \)
79 \( 1 + (-0.169 - 0.0980i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.96 - 8.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.19 - 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−9.724667354900275850984228861083, −8.844157754371529533515450826089, −7.72091566870909207429672893814, −7.10291904435687539948480390299, −6.33766179214080872604898434387, −5.18074505051056525125209611008, −4.25043746480939575025856416942, −3.41589326525930644051298024250, −2.06865745465827218817956831358, −1.10702026584507722750464599558, 1.97599574757692054837633050136, 2.76449855482346181568430756024, 4.23390965597585915922729339663, 4.81981029267178960552202250848, 5.80482597388852968616672723722, 6.37613647204296684491145733113, 7.72702665898368536917818090295, 8.523817565852962375295648902382, 9.050004117741158666827168718926, 9.916910322655782158353552091135

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