Properties

Label 2-5600-1.1-c1-0-31
Degree $2$
Conductor $5600$
Sign $1$
Analytic cond. $44.7162$
Root an. cond. $6.68701$
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  − 1.83·3-s + 7-s + 0.359·9-s + 4.40·11-s + 3.20·13-s − 1.14·17-s − 1.72·19-s − 1.83·21-s + 3.25·23-s + 4.83·27-s + 4.18·29-s + 1.36·31-s − 8.06·33-s − 4.19·37-s − 5.88·39-s + 11.7·41-s − 2.64·43-s + 0.106·47-s + 49-s + 2.10·51-s − 7.86·53-s + 3.16·57-s + 13.3·59-s − 10.0·61-s + 0.359·63-s + 1.28·67-s − 5.96·69-s + ⋯
L(s)  = 1  − 1.05·3-s + 0.377·7-s + 0.119·9-s + 1.32·11-s + 0.889·13-s − 0.278·17-s − 0.395·19-s − 0.399·21-s + 0.678·23-s + 0.931·27-s + 0.777·29-s + 0.245·31-s − 1.40·33-s − 0.690·37-s − 0.941·39-s + 1.83·41-s − 0.404·43-s + 0.0155·47-s + 0.142·49-s + 0.294·51-s − 1.08·53-s + 0.419·57-s + 1.73·59-s − 1.28·61-s + 0.0453·63-s + 0.156·67-s − 0.717·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(44.7162\)
Root analytic conductor: \(6.68701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542604637\)
\(L(\frac12)\) \(\approx\) \(1.542604637\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + 1.83T + 3T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
19 \( 1 + 1.72T + 19T^{2} \)
23 \( 1 - 3.25T + 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 - 1.36T + 31T^{2} \)
37 \( 1 + 4.19T + 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 2.64T + 43T^{2} \)
47 \( 1 - 0.106T + 47T^{2} \)
53 \( 1 + 7.86T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 1.28T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 - 5.48T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 - 4.31T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−8.283994115145815512292113566851, −7.20944233214373357840736804319, −6.50717729387408318569690189377, −6.12504909026825829938338176991, −5.31518726839905365748484693949, −4.53844023141174070610623669582, −3.88496143879875476406589089924, −2.84162258515240097357752685420, −1.56735090245609113726275050633, −0.75535343310244052859004847512, 0.75535343310244052859004847512, 1.56735090245609113726275050633, 2.84162258515240097357752685420, 3.88496143879875476406589089924, 4.53844023141174070610623669582, 5.31518726839905365748484693949, 6.12504909026825829938338176991, 6.50717729387408318569690189377, 7.20944233214373357840736804319, 8.283994115145815512292113566851

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