Properties

Label 2-1110-1.1-c1-0-3
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 + 2.37·7-s − 8-s + 9-s − 10-s + 4.37·11-s − 12-s − 4.37·13-s − 2.37·14-s − 15-s + 16-s + 4.37·17-s − 18-s + 4.37·19-s + 20-s − 2.37·21-s − 4.37·22-s − 2.37·23-s + 24-s + 25-s + 4.37·26-s − 27-s + 2.37·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.896·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.31·11-s − 0.288·12-s − 1.21·13-s − 0.634·14-s − 0.258·15-s + 0.250·16-s + 1.06·17-s − 0.235·18-s + 1.00·19-s + 0.223·20-s − 0.517·21-s − 0.932·22-s − 0.494·23-s + 0.204·24-s + 0.200·25-s + 0.857·26-s − 0.192·27-s + 0.448·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.247897075\)
\(L(\frac12)\) \(\approx\) \(1.247897075\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 + 4.37T + 13T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 - 4.37T + 19T^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8.74T + 31T^{2} \)
41 \( 1 + 2.74T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 + 0.372T + 53T^{2} \)
59 \( 1 - 3.25T + 59T^{2} \)
61 \( 1 - 8.74T + 61T^{2} \)
67 \( 1 + 8.74T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6.37T + 83T^{2} \)
89 \( 1 - 7.62T + 89T^{2} \)
97 \( 1 + 9.48T + 97T^{2} \)
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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

−9.785166473869818352386352040370, −9.233962082308994120919370484186, −8.166292991905687590382797175022, −7.37109103034187218877082252533, −6.64367677864087529475494925552, −5.58163714663183829670662499015, −4.89680318143745927342182143807, −3.57733667604527650710703646354, −2.06079048662367972003274439679, −1.04556555540668435101019840246, 1.04556555540668435101019840246, 2.06079048662367972003274439679, 3.57733667604527650710703646354, 4.89680318143745927342182143807, 5.58163714663183829670662499015, 6.64367677864087529475494925552, 7.37109103034187218877082252533, 8.166292991905687590382797175022, 9.233962082308994120919370484186, 9.785166473869818352386352040370

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