Properties

Label 2-644-1.1-c1-0-0
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  − 1.76·3-s − 3.11·5-s − 7-s + 0.108·9-s − 5.23·11-s + 6.34·13-s + 5.49·15-s − 0.585·17-s + 8.48·19-s + 1.76·21-s − 23-s + 4.71·25-s + 5.09·27-s − 1.59·29-s − 5.97·31-s + 9.23·33-s + 3.11·35-s + 10.0·37-s − 11.1·39-s + 1.17·41-s − 2.96·43-s − 0.339·45-s − 2.55·47-s + 49-s + 1.03·51-s + 8.27·53-s + 16.3·55-s + ⋯
L(s)  = 1  − 1.01·3-s − 1.39·5-s − 0.377·7-s + 0.0363·9-s − 1.57·11-s + 1.76·13-s + 1.41·15-s − 0.141·17-s + 1.94·19-s + 0.384·21-s − 0.208·23-s + 0.942·25-s + 0.981·27-s − 0.296·29-s − 1.07·31-s + 1.60·33-s + 0.526·35-s + 1.64·37-s − 1.79·39-s + 0.183·41-s − 0.451·43-s − 0.0505·45-s − 0.372·47-s + 0.142·49-s + 0.144·51-s + 1.13·53-s + 2.20·55-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

\(L(1)\) \(\approx\) \(0.5959991900\)
\(L(\frac12)\) \(\approx\) \(0.5959991900\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.76T + 3T^{2} \)
5 \( 1 + 3.11T + 5T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 - 6.34T + 13T^{2} \)
17 \( 1 + 0.585T + 17T^{2} \)
19 \( 1 - 8.48T + 19T^{2} \)
29 \( 1 + 1.59T + 29T^{2} \)
31 \( 1 + 5.97T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 1.17T + 41T^{2} \)
43 \( 1 + 2.96T + 43T^{2} \)
47 \( 1 + 2.55T + 47T^{2} \)
53 \( 1 - 8.27T + 53T^{2} \)
59 \( 1 - 14.6T + 59T^{2} \)
61 \( 1 - 6.15T + 61T^{2} \)
67 \( 1 + 8.52T + 67T^{2} \)
71 \( 1 - 8.38T + 71T^{2} \)
73 \( 1 + 0.358T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 + 8.38T + 97T^{2} \)
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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

−10.92947006105620566658346881537, −9.916749618906156769477401401151, −8.629577424279488764367254660426, −7.87737309144978461153807221415, −7.08700900158142949064811047588, −5.86592199269953802002697298053, −5.25793256535673690117212085756, −3.97100658097703822156604634605, −3.04340842656167423704613271672, −0.68321297020692240351439361056, 0.68321297020692240351439361056, 3.04340842656167423704613271672, 3.97100658097703822156604634605, 5.25793256535673690117212085756, 5.86592199269953802002697298053, 7.08700900158142949064811047588, 7.87737309144978461153807221415, 8.629577424279488764367254660426, 9.916749618906156769477401401151, 10.92947006105620566658346881537

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