Properties

Label 2-1110-1.1-c1-0-21
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 6.12·11-s − 12-s + 5.68·13-s − 14-s + 15-s + 16-s − 5·17-s + 18-s + 0.561·19-s − 20-s + 21-s − 6.12·22-s − 6.56·23-s − 24-s + 25-s + 5.68·26-s − 27-s − 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 − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.84·11-s − 0.288·12-s + 1.57·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s + 0.128·19-s − 0.223·20-s + 0.218·21-s − 1.30·22-s − 1.36·23-s − 0.204·24-s + 0.200·25-s + 1.11·26-s − 0.192·27-s − 0.188·28-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: \(1\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 7T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 - 5.68T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 0.561T + 19T^{2} \)
23 \( 1 + 6.56T + 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 + 3.56T + 31T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 5.24T + 53T^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 1.12T + 71T^{2} \)
73 \( 1 + 6.56T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 7.43T + 83T^{2} \)
89 \( 1 - 5.68T + 89T^{2} \)
97 \( 1 - 3.31T + 97T^{2} \)
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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

−9.600516873136888235247570747972, −8.369103931345263489013047968798, −7.73589678570944847890062228996, −6.68809362990095016628647885460, −5.93895383136299238805627461161, −5.17864061466260350396003325547, −4.16743143517120259529648606362, −3.30027531686472465337447397895, −1.99602686798888578561163090851, 0, 1.99602686798888578561163090851, 3.30027531686472465337447397895, 4.16743143517120259529648606362, 5.17864061466260350396003325547, 5.93895383136299238805627461161, 6.68809362990095016628647885460, 7.73589678570944847890062228996, 8.369103931345263489013047968798, 9.600516873136888235247570747972

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