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Properties

Label 2-1110-185.159-c1-0-31
Degree $2$
Conductor $1110$
Sign $-0.845 + 0.534i$
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 + (1.05 + 1.96i)5-s − 0.999i·6-s + (−3.29 − 1.90i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (1.17 − 1.90i)10-s − 3.84·11-s + (−0.866 + 0.499i)12-s + (1.22 − 2.12i)13-s + 3.80i·14-s + (−0.0669 + 2.23i)15-s + (−0.5 − 0.866i)16-s + (−3.42 − 5.93i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.473 + 0.880i)5-s − 0.408i·6-s + (−1.24 − 0.718i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.371 − 0.601i)10-s − 1.16·11-s + (−0.249 + 0.144i)12-s + (0.339 − 0.588i)13-s + 1.01i·14-s + (−0.0172 + 0.577i)15-s + (−0.125 − 0.216i)16-s + (−0.831 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.845 + 0.534i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.845 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5624175797\)
\(L(\frac12)\) \(\approx\) \(0.5624175797\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 - 1.96i)T \)
37 \( 1 + (-4.13 + 4.46i)T \)
good7 \( 1 + (3.29 + 1.90i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.84T + 11T^{2} \)
13 \( 1 + (-1.22 + 2.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.42 + 5.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.996 + 0.575i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.0656T + 23T^{2} \)
29 \( 1 - 2.70iT - 29T^{2} \)
31 \( 1 + 8.12iT - 31T^{2} \)
41 \( 1 + (-0.239 + 0.415i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 1.94iT - 47T^{2} \)
53 \( 1 + (-4.83 + 2.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.87 + 1.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.07 + 2.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.70 + 3.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.72 - 6.44i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.82iT - 73T^{2} \)
79 \( 1 + (-2.57 - 1.48i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.83 + 2.79i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.89 + 3.98i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.90T + 97T^{2} \)
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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

−9.686983099531940033756939876752, −9.007845725623462272726084822129, −7.81685037190452163675463222461, −7.17709052371651805936418205432, −6.28395016927738017904942400547, −5.07573205036142966850678256068, −3.80320028269145233993764667533, −2.98486226785687998535090503294, −2.35399744689749040260937396706, −0.24403103589138299602402203729, 1.65286100892657334150833526632, 2.79000079569503026998411701129, 4.15234118202279608476138204644, 5.27477988980504196212268821645, 6.17853847874233207237191365137, 6.65596785741149789393427047111, 7.962088307751293204773059658539, 8.605168562616892073337087733984, 9.099172414715419281432675166436, 9.944397790240271078508223763538

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