Properties

Label 2-5586-1.1-c1-0-2
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 − 2.72·5-s + 6-s − 8-s + 9-s + 2.72·10-s + 3.36·11-s − 12-s − 6.56·13-s + 2.72·15-s + 16-s − 0.683·17-s − 18-s + 19-s − 2.72·20-s − 3.36·22-s + 1.07·23-s + 24-s + 2.40·25-s + 6.56·26-s − 27-s − 1.51·29-s − 2.72·30-s + 0.107·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.21·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.860·10-s + 1.01·11-s − 0.288·12-s − 1.82·13-s + 0.702·15-s + 0.250·16-s − 0.165·17-s − 0.235·18-s + 0.229·19-s − 0.608·20-s − 0.717·22-s + 0.224·23-s + 0.204·24-s + 0.481·25-s + 1.28·26-s − 0.192·27-s − 0.280·29-s − 0.496·30-s + 0.0192·31-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5586,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4584871365\)
\(L(\frac12)\) \(\approx\) \(0.4584871365\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.72T + 5T^{2} \)
11 \( 1 - 3.36T + 11T^{2} \)
13 \( 1 + 6.56T + 13T^{2} \)
17 \( 1 + 0.683T + 17T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 + 1.51T + 29T^{2} \)
31 \( 1 - 0.107T + 31T^{2} \)
37 \( 1 - 6.09T + 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 + 2.23T + 43T^{2} \)
47 \( 1 + 8.70T + 47T^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 7.84T + 61T^{2} \)
67 \( 1 + 8.33T + 67T^{2} \)
71 \( 1 + 4.27T + 71T^{2} \)
73 \( 1 - 8.19T + 73T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 + 1.95T + 83T^{2} \)
89 \( 1 - 7.93T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−7.898160250219170156883842188210, −7.58314314357487047572297207483, −6.84645452898415061572127101652, −6.28073761188773088014572858942, −5.14962064001771034721913691318, −4.53064786825550355599968527995, −3.70953001275283046813336243955, −2.76918094261223001464796394918, −1.61150140686520767782156114051, −0.41679327508158865646351975146, 0.41679327508158865646351975146, 1.61150140686520767782156114051, 2.76918094261223001464796394918, 3.70953001275283046813336243955, 4.53064786825550355599968527995, 5.14962064001771034721913691318, 6.28073761188773088014572858942, 6.84645452898415061572127101652, 7.58314314357487047572297207483, 7.898160250219170156883842188210

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