Properties

Label 2-5586-1.1-c1-0-18
Degree $2$
Conductor $5586$
Sign $1$
Analytic cond. $44.6044$
Root an. cond. $6.67865$
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 − 3.46·5-s + 6-s − 8-s + 9-s + 3.46·10-s + 0.278·11-s − 12-s + 6.92·13-s + 3.46·15-s + 16-s + 5.79·17-s − 18-s + 19-s − 3.46·20-s − 0.278·22-s + 3.09·23-s + 24-s + 6.98·25-s − 6.92·26-s − 27-s − 8.55·29-s − 3.46·30-s + 7.34·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.54·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.0838·11-s − 0.288·12-s + 1.91·13-s + 0.893·15-s + 0.250·16-s + 1.40·17-s − 0.235·18-s + 0.229·19-s − 0.773·20-s − 0.0593·22-s + 0.644·23-s + 0.204·24-s + 1.39·25-s − 1.35·26-s − 0.192·27-s − 1.58·29-s − 0.631·30-s + 1.32·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5586\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(44.6044\)
Root analytic conductor: \(6.67865\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5586} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5586,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9575941405\)
\(L(\frac12)\) \(\approx\) \(0.9575941405\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 - 0.278T + 11T^{2} \)
13 \( 1 - 6.92T + 13T^{2} \)
17 \( 1 - 5.79T + 17T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 + 8.55T + 29T^{2} \)
31 \( 1 - 7.34T + 31T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 - 8.88T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 - 8.38T + 53T^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 - 0.684T + 71T^{2} \)
73 \( 1 + 1.92T + 73T^{2} \)
79 \( 1 - 1.40T + 79T^{2} \)
83 \( 1 + 0.146T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 4.09T + 97T^{2} \)
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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

−8.027154816023088368330648636145, −7.58305996509890733349703829200, −6.93108621189023595965324681089, −6.02018221672547255851959453500, −5.47462493426065146735221176690, −4.25828589944899078912211299922, −3.72127727765906776934097025754, −2.97045715824663710751863616726, −1.35122825452433132514110915833, −0.68652414094213973213125007465, 0.68652414094213973213125007465, 1.35122825452433132514110915833, 2.97045715824663710751863616726, 3.72127727765906776934097025754, 4.25828589944899078912211299922, 5.47462493426065146735221176690, 6.02018221672547255851959453500, 6.93108621189023595965324681089, 7.58305996509890733349703829200, 8.027154816023088368330648636145

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