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Properties

Label 2-1110-37.11-c1-0-21
Degree $2$
Conductor $1110$
Sign $-0.766 + 0.641i$
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 + (−1.55 + 2.68i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s − 1.08·11-s + (−0.499 − 0.866i)12-s + (−1.53 − 0.885i)13-s − 3.10i·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−4.39 + 2.53i)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 + (−0.586 + 1.01i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s − 0.326·11-s + (−0.144 − 0.249i)12-s + (−0.425 − 0.245i)13-s − 0.829i·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (−1.06 + 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.766 + 0.641i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.766 + 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2860844342\)
\(L(\frac12)\) \(\approx\) \(0.2860844342\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.59 - 2.38i)T \)
good7 \( 1 + (1.55 - 2.68i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 1.08T + 11T^{2} \)
13 \( 1 + (1.53 + 0.885i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.39 - 2.53i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.08 + 3.51i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.39iT - 23T^{2} \)
29 \( 1 - 2.60iT - 29T^{2} \)
31 \( 1 + 9.58iT - 31T^{2} \)
41 \( 1 + (-5.39 + 9.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 0.223iT - 43T^{2} \)
47 \( 1 - 6.17T + 47T^{2} \)
53 \( 1 + (1.31 + 2.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.47 + 0.851i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.59 + 3.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.526 - 0.911i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.27 - 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + (6.89 + 3.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.32 - 14.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-11.8 + 6.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−9.120473651264571188339759020131, −8.869455699587937442096123098852, −8.005853917330497467515241970866, −6.89356833897605371025911374377, −6.37052010113550609154231350607, −5.59993218760803594536735360334, −4.32133192107531648743034280833, −2.58460889797520014812726323551, −2.22475966277349038886326605904, −0.13587443000573074363298483281, 1.66219377233429657729413783067, 2.89021672932701798430955096730, 3.92302507755164167567738521242, 4.76618093752123431461020721941, 6.05907638429648030320088457218, 7.03866246091880074728869592165, 7.74599593733062991171886499216, 8.840822919127839227968569820484, 9.330867639233358907663958430433, 10.27060624343547501507681712535

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