Properties

Label 2-1110-1.1-c1-0-8
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 + 3·7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 6.81·13-s − 3·14-s + 15-s + 16-s + 17-s − 18-s − 2.81·19-s + 20-s + 3·21-s + 22-s − 4.81·23-s − 24-s + 25-s − 6.81·26-s + 27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.89·13-s − 0.801·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 0.645·19-s + 0.223·20-s + 0.654·21-s + 0.213·22-s − 1.00·23-s − 0.204·24-s + 0.200·25-s − 1.33·26-s + 0.192·27-s + 0.566·28-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

\(L(1)\) \(\approx\) \(1.879342531\)
\(L(\frac12)\) \(\approx\) \(1.879342531\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 6.81T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + 2.81T + 19T^{2} \)
23 \( 1 + 4.81T + 23T^{2} \)
29 \( 1 - 3.81T + 29T^{2} \)
31 \( 1 + 3.81T + 31T^{2} \)
41 \( 1 - 5.81T + 41T^{2} \)
43 \( 1 - 5.81T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 + 3.81T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 9.63T + 71T^{2} \)
73 \( 1 - 4.81T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 6.81T + 83T^{2} \)
89 \( 1 - 1.18T + 89T^{2} \)
97 \( 1 - 5.81T + 97T^{2} \)
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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

−9.699398934739523573265107272458, −8.918654842116913631225753474524, −8.185197986186436736572992590711, −7.81154950148644274900401827546, −6.51559226175132754994952893089, −5.80244498102948153256331981297, −4.56616498460899275567954514308, −3.47361760455742440863304118553, −2.15955810909539091457960754382, −1.29179280352691421808205394218, 1.29179280352691421808205394218, 2.15955810909539091457960754382, 3.47361760455742440863304118553, 4.56616498460899275567954514308, 5.80244498102948153256331981297, 6.51559226175132754994952893089, 7.81154950148644274900401827546, 8.185197986186436736572992590711, 8.918654842116913631225753474524, 9.699398934739523573265107272458

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