Properties

Label 2-1110-37.10-c1-0-2
Degree $2$
Conductor $1110$
Sign $-0.948 + 0.316i$
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 + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (−0.210 − 0.364i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s − 3.22·11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)13-s + 0.421·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−1.28 + 2.23i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (−0.0796 − 0.137i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s − 0.971·11-s + (0.144 − 0.249i)12-s + (−0.138 − 0.240i)13-s + 0.112·14-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.312 + 0.541i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6452040785\)
\(L(\frac12)\) \(\approx\) \(0.6452040785\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (2.07 - 5.71i)T \)
good7 \( 1 + (0.210 + 0.364i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.22T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.28 - 2.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.11 - 3.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.60T + 23T^{2} \)
29 \( 1 + 7.24T + 29T^{2} \)
31 \( 1 + 2.44T + 31T^{2} \)
41 \( 1 + (-5.12 - 8.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.842T + 43T^{2} \)
47 \( 1 + 8.82T + 47T^{2} \)
53 \( 1 + (-1.32 + 2.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.03 + 1.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.900 - 1.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.93 - 3.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.58 - 6.21i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.44T + 73T^{2} \)
79 \( 1 + (7.37 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.01 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.42 - 5.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.00063173005227696191315094867, −9.695245988738691956403243110582, −8.468788282654308975462801576023, −7.934930856426963869314644929642, −7.11291145660324562225490788406, −6.00214181799484897718108805663, −5.38668946527403644271120749970, −4.24522968139683531004663369749, −3.19842357936745317104341345196, −1.90086797633417812248260223145, 0.28704669500756113110904083768, 1.88952125598600164146787691791, 2.68289265648492365211391702278, 3.89081144611123849016734245845, 5.05322786047300883353350854386, 5.92709532320300527795502099489, 7.24917390472151777028204711578, 7.71596733874737497328443068962, 8.805526623324744967841700779569, 9.278226247460945045268698283985

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