Properties

Label 2-644-1.1-c1-0-8
Degree $2$
Conductor $644$
Sign $1$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
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  + 2.18·3-s + 4.04·5-s − 7-s + 1.76·9-s − 3.98·11-s + 6.74·13-s + 8.82·15-s − 4.92·17-s − 1.24·19-s − 2.18·21-s − 23-s + 11.3·25-s − 2.70·27-s + 6.48·29-s − 8.38·31-s − 8.69·33-s − 4.04·35-s − 7.61·37-s + 14.7·39-s − 7.11·41-s − 1.11·43-s + 7.12·45-s + 8.31·47-s + 49-s − 10.7·51-s − 4.77·53-s − 16.1·55-s + ⋯
L(s)  = 1  + 1.25·3-s + 1.80·5-s − 0.377·7-s + 0.587·9-s − 1.20·11-s + 1.87·13-s + 2.27·15-s − 1.19·17-s − 0.286·19-s − 0.476·21-s − 0.208·23-s + 2.26·25-s − 0.519·27-s + 1.20·29-s − 1.50·31-s − 1.51·33-s − 0.683·35-s − 1.25·37-s + 2.35·39-s − 1.11·41-s − 0.170·43-s + 1.06·45-s + 1.21·47-s + 0.142·49-s − 1.50·51-s − 0.655·53-s − 2.17·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.727161199\)
\(L(\frac12)\) \(\approx\) \(2.727161199\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good3 \( 1 - 2.18T + 3T^{2} \)
5 \( 1 - 4.04T + 5T^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 + 4.92T + 17T^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
29 \( 1 - 6.48T + 29T^{2} \)
31 \( 1 + 8.38T + 31T^{2} \)
37 \( 1 + 7.61T + 37T^{2} \)
41 \( 1 + 7.11T + 41T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 - 8.31T + 47T^{2} \)
53 \( 1 + 4.77T + 53T^{2} \)
59 \( 1 - 1.31T + 59T^{2} \)
61 \( 1 - 0.838T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 6.46T + 73T^{2} \)
79 \( 1 + 6.45T + 79T^{2} \)
83 \( 1 + 5.59T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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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

−10.42029542754967443033398894814, −9.574961492571575205232757838248, −8.740698215727838775750107892871, −8.405488233453608066699893497362, −6.92202312760197568127741131781, −6.09288142992935337203432906932, −5.21424969637098265872416757539, −3.65349872272827095655780785260, −2.60411991707496156001830270983, −1.79632357507754833535704374709, 1.79632357507754833535704374709, 2.60411991707496156001830270983, 3.65349872272827095655780785260, 5.21424969637098265872416757539, 6.09288142992935337203432906932, 6.92202312760197568127741131781, 8.405488233453608066699893497362, 8.740698215727838775750107892871, 9.574961492571575205232757838248, 10.42029542754967443033398894814

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