Properties

Label 2-1110-1.1-c1-0-18
Degree $2$
Conductor $1110$
Sign $1$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 5.13·7-s + 8-s + 9-s + 10-s + 2.22·11-s + 12-s − 6.60·13-s + 5.13·14-s + 15-s + 16-s + 5.63·17-s + 18-s − 7.51·19-s + 20-s + 5.13·21-s + 2.22·22-s − 6.10·23-s + 24-s + 25-s − 6.60·26-s + 27-s + 5.13·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.94·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.669·11-s + 0.288·12-s − 1.83·13-s + 1.37·14-s + 0.258·15-s + 0.250·16-s + 1.36·17-s + 0.235·18-s − 1.72·19-s + 0.223·20-s + 1.12·21-s + 0.473·22-s − 1.27·23-s + 0.204·24-s + 0.200·25-s − 1.29·26-s + 0.192·27-s + 0.970·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.799593101\)
\(L(\frac12)\) \(\approx\) \(3.799593101\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
37 \( 1 + T \)
good7 \( 1 - 5.13T + 7T^{2} \)
11 \( 1 - 2.22T + 11T^{2} \)
13 \( 1 + 6.60T + 13T^{2} \)
17 \( 1 - 5.63T + 17T^{2} \)
19 \( 1 + 7.51T + 19T^{2} \)
23 \( 1 + 6.10T + 23T^{2} \)
29 \( 1 + 7.29T + 29T^{2} \)
31 \( 1 + 0.967T + 31T^{2} \)
41 \( 1 + 1.03T + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 + 5.85T + 47T^{2} \)
53 \( 1 + 9.63T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 - 1.49T + 67T^{2} \)
71 \( 1 - 0.329T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 9.04T + 83T^{2} \)
89 \( 1 + 2.27T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−9.903600277335397607697018104168, −8.994595896041201859743668063080, −7.84641291040039046076858330195, −7.66756311934653492188336425223, −6.39276051051326307062073783110, −5.31704396473029676438462091860, −4.67205808356557498232867444452, −3.80199104308531425991935849404, −2.28095104184203708801932837653, −1.73599861733114192751296213316, 1.73599861733114192751296213316, 2.28095104184203708801932837653, 3.80199104308531425991935849404, 4.67205808356557498232867444452, 5.31704396473029676438462091860, 6.39276051051326307062073783110, 7.66756311934653492188336425223, 7.84641291040039046076858330195, 8.994595896041201859743668063080, 9.903600277335397607697018104168

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