Properties

Label 2-644-1.1-c1-0-9
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  + 3.09·3-s + 1.68·5-s + 7-s + 6.56·9-s + 1.03·11-s − 4.59·13-s + 5.21·15-s − 1.50·17-s − 7.62·19-s + 3.09·21-s + 23-s − 2.15·25-s + 11.0·27-s − 1.99·29-s − 6.28·31-s + 3.19·33-s + 1.68·35-s + 7.43·37-s − 14.2·39-s + 2.41·41-s + 9.75·43-s + 11.0·45-s − 5.47·47-s + 49-s − 4.65·51-s − 14.0·53-s + 1.74·55-s + ⋯
L(s)  = 1  + 1.78·3-s + 0.754·5-s + 0.377·7-s + 2.18·9-s + 0.311·11-s − 1.27·13-s + 1.34·15-s − 0.365·17-s − 1.74·19-s + 0.674·21-s + 0.208·23-s − 0.430·25-s + 2.12·27-s − 0.370·29-s − 1.12·31-s + 0.555·33-s + 0.285·35-s + 1.22·37-s − 2.27·39-s + 0.376·41-s + 1.48·43-s + 1.65·45-s − 0.798·47-s + 0.142·49-s − 0.652·51-s − 1.92·53-s + 0.234·55-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

\(L(1)\) \(\approx\) \(2.939229813\)
\(L(\frac12)\) \(\approx\) \(2.939229813\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.09T + 3T^{2} \)
5 \( 1 - 1.68T + 5T^{2} \)
11 \( 1 - 1.03T + 11T^{2} \)
13 \( 1 + 4.59T + 13T^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + 7.62T + 19T^{2} \)
29 \( 1 + 1.99T + 29T^{2} \)
31 \( 1 + 6.28T + 31T^{2} \)
37 \( 1 - 7.43T + 37T^{2} \)
41 \( 1 - 2.41T + 41T^{2} \)
43 \( 1 - 9.75T + 43T^{2} \)
47 \( 1 + 5.47T + 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 0.248T + 61T^{2} \)
67 \( 1 - 6.60T + 67T^{2} \)
71 \( 1 + 3.58T + 71T^{2} \)
73 \( 1 - 2.21T + 73T^{2} \)
79 \( 1 - 5.84T + 79T^{2} \)
83 \( 1 + 7.50T + 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−10.23054556414928730682674972284, −9.463451754443065556075853560327, −8.965601261938094561598799435561, −8.014447996932126343988249518933, −7.31425172924259722510601788242, −6.22704745559759103167544381516, −4.76790115555116901445658646274, −3.86176491088755739329487811105, −2.49072573921677739459001514456, −1.93304379953186733955596165812, 1.93304379953186733955596165812, 2.49072573921677739459001514456, 3.86176491088755739329487811105, 4.76790115555116901445658646274, 6.22704745559759103167544381516, 7.31425172924259722510601788242, 8.014447996932126343988249518933, 8.965601261938094561598799435561, 9.463451754443065556075853560327, 10.23054556414928730682674972284

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