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Properties

Label 2-1110-185.84-c1-0-1
Degree $2$
Conductor $1110$
Sign $-0.974 + 0.225i$
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 + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−2.13 + 0.662i)5-s + 0.999·6-s + (−1.09 + 0.631i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.18 − 0.494i)10-s − 4.99·11-s + (0.866 + 0.499i)12-s + (−4.64 + 2.68i)13-s − 1.26·14-s + (−1.51 + 1.64i)15-s + (−0.5 + 0.866i)16-s + (3.69 + 2.13i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.955 + 0.296i)5-s + 0.408·6-s + (−0.413 + 0.238i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.689 − 0.156i)10-s − 1.50·11-s + (0.249 + 0.144i)12-s + (−1.28 + 0.743i)13-s − 0.337·14-s + (−0.392 + 0.423i)15-s + (−0.125 + 0.216i)16-s + (0.895 + 0.516i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3028541366\)
\(L(\frac12)\) \(\approx\) \(0.3028541366\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (2.13 - 0.662i)T \)
37 \( 1 + (5.93 + 1.32i)T \)
good7 \( 1 + (1.09 - 0.631i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 + (4.64 - 2.68i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.69 - 2.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.67 + 6.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.94iT - 23T^{2} \)
29 \( 1 - 0.263T + 29T^{2} \)
31 \( 1 + 8.35T + 31T^{2} \)
41 \( 1 + (-3.74 - 6.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 - 3.66iT - 47T^{2} \)
53 \( 1 + (-7.06 - 4.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.45 - 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.390 - 0.675i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.05 - 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.141 + 0.244i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.54iT - 73T^{2} \)
79 \( 1 + (4.05 + 7.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.9 - 6.88i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.52 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.02iT - 97T^{2} \)
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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

−10.42763908895111224992060298745, −9.285859365070035663749608760569, −8.401497732362754359653755698120, −7.64044408944300584773280414272, −7.09025580238798214441702865493, −6.21536666701809008326987609105, −4.94430453155381156822718380451, −4.29308720280700446465284162571, −2.97671612839949655643994348075, −2.47036104871031520616734867307, 0.092008091964278216534875583271, 2.12645386725351624178681813006, 3.34488848050721691508542608934, 3.78477468634158097058671242742, 5.21610722729259664420152050156, 5.41466246984735353309700359999, 7.27946761959436037151416233023, 7.58447878901746057221480010722, 8.479276905240409347382471037409, 9.660041843156932358175460458864

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