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Properties

Label 2-1110-37.11-c1-0-0
Degree $2$
Conductor $1110$
Sign $-0.944 + 0.327i$
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 + (0.844 − 1.46i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 3.20·11-s + (0.499 + 0.866i)12-s + (3.91 + 2.26i)13-s + 1.68i·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−2.32 + 1.34i)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 + (0.319 − 0.553i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 0.965·11-s + (0.144 + 0.249i)12-s + (1.08 + 0.627i)13-s + 0.451i·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.564 + 0.326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.944 + 0.327i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09006156122\)
\(L(\frac12)\) \(\approx\) \(0.09006156122\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.07 - 0.237i)T \)
good7 \( 1 + (-0.844 + 1.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.20T + 11T^{2} \)
13 \( 1 + (-3.91 - 2.26i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.32 - 1.34i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.29 + 1.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.25iT - 23T^{2} \)
29 \( 1 - 6.29iT - 29T^{2} \)
31 \( 1 - 2.13iT - 31T^{2} \)
41 \( 1 + (5.65 - 9.78i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.12iT - 43T^{2} \)
47 \( 1 + 8.83T + 47T^{2} \)
53 \( 1 + (6.84 + 11.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.611 + 0.352i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.81 + 5.08i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.212 - 0.367i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.72 + 6.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + (-3.03 - 1.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.09 - 5.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.23 + 0.710i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−10.45942527591398934790215923138, −9.403730548201563412911010067674, −8.478504445522370168903502375725, −8.149915266956762559056680889733, −6.87622134463958837925955834138, −6.33454602962637800345643577983, −5.03469203859666922909858237877, −4.44777446852670194479202468922, −3.21146727522679967046008094904, −1.57262685467464723290722826836, 0.05015036711105207806542430430, 1.68824892454472151296082299957, 2.76142356100917998388575707758, 3.86624669243664661535376496438, 5.25509727735260744592247048427, 6.02970806870266608793679590777, 7.08916722717825470672484301238, 7.917756995767689714637399167116, 8.419063684369893702451256375991, 9.314541761686748656607519785637

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