Properties

Label 2-5610-1.1-c1-0-47
Degree $2$
Conductor $5610$
Sign $1$
Analytic cond. $44.7960$
Root an. cond. $6.69298$
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 + 1.15·7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 0.842·13-s + 1.15·14-s − 15-s + 16-s + 17-s + 18-s + 5.15·19-s − 20-s + 1.15·21-s − 22-s − 2.84·23-s + 24-s + 25-s + 0.842·26-s + 27-s + 1.15·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.437·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.233·13-s + 0.309·14-s − 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.18·19-s − 0.223·20-s + 0.252·21-s − 0.213·22-s − 0.592·23-s + 0.204·24-s + 0.200·25-s + 0.165·26-s + 0.192·27-s + 0.218·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(44.7960\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.128502847\)
\(L(\frac12)\) \(\approx\) \(4.128502847\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 - 1.15T + 7T^{2} \)
13 \( 1 - 0.842T + 13T^{2} \)
19 \( 1 - 5.15T + 19T^{2} \)
23 \( 1 + 2.84T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 8.34T + 31T^{2} \)
37 \( 1 - 8.65T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 5.50T + 43T^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 5.50T + 59T^{2} \)
61 \( 1 + 4.65T + 61T^{2} \)
67 \( 1 - 1.34T + 67T^{2} \)
71 \( 1 + 7.81T + 71T^{2} \)
73 \( 1 - 9.18T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 0.343T + 83T^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 - 8.02T + 97T^{2} \)
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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

−7.973083262779192032256357885736, −7.60771328888713831063891614763, −6.65889073274200544270750598425, −6.00688541238882429880225040201, −4.96972523711986785022394533978, −4.58444377535302019798648341882, −3.55702019419960767748169858130, −3.04543220211847221757082554308, −2.06954568093964844056573119252, −0.989786910029940653778111025198, 0.989786910029940653778111025198, 2.06954568093964844056573119252, 3.04543220211847221757082554308, 3.55702019419960767748169858130, 4.58444377535302019798648341882, 4.96972523711986785022394533978, 6.00688541238882429880225040201, 6.65889073274200544270750598425, 7.60771328888713831063891614763, 7.973083262779192032256357885736

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