Properties

Label 2-644-1.1-c1-0-7
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  + 1.43·3-s + 2.34·5-s − 7-s − 0.938·9-s + 5.38·11-s − 5.32·13-s + 3.36·15-s + 7.88·17-s + 6.06·19-s − 1.43·21-s − 23-s + 0.486·25-s − 5.65·27-s + 1.87·29-s − 5.83·31-s + 7.73·33-s − 2.34·35-s + 1.19·37-s − 7.64·39-s + 6.45·41-s − 6.93·43-s − 2.19·45-s − 10.8·47-s + 49-s + 11.3·51-s + 7.94·53-s + 12.6·55-s + ⋯
L(s)  = 1  + 0.828·3-s + 1.04·5-s − 0.377·7-s − 0.312·9-s + 1.62·11-s − 1.47·13-s + 0.868·15-s + 1.91·17-s + 1.39·19-s − 0.313·21-s − 0.208·23-s + 0.0973·25-s − 1.08·27-s + 0.348·29-s − 1.04·31-s + 1.34·33-s − 0.395·35-s + 0.195·37-s − 1.22·39-s + 1.00·41-s − 1.05·43-s − 0.327·45-s − 1.58·47-s + 0.142·49-s + 1.58·51-s + 1.09·53-s + 1.70·55-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

\(L(1)\) \(\approx\) \(2.276075473\)
\(L(\frac12)\) \(\approx\) \(2.276075473\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.43T + 3T^{2} \)
5 \( 1 - 2.34T + 5T^{2} \)
11 \( 1 - 5.38T + 11T^{2} \)
13 \( 1 + 5.32T + 13T^{2} \)
17 \( 1 - 7.88T + 17T^{2} \)
19 \( 1 - 6.06T + 19T^{2} \)
29 \( 1 - 1.87T + 29T^{2} \)
31 \( 1 + 5.83T + 31T^{2} \)
37 \( 1 - 1.19T + 37T^{2} \)
41 \( 1 - 6.45T + 41T^{2} \)
43 \( 1 + 6.93T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 7.94T + 53T^{2} \)
59 \( 1 + 2.69T + 59T^{2} \)
61 \( 1 + 3.27T + 61T^{2} \)
67 \( 1 + 1.82T + 67T^{2} \)
71 \( 1 - 7.30T + 71T^{2} \)
73 \( 1 + 7.51T + 73T^{2} \)
79 \( 1 + 2.49T + 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−10.07988906742863991427306989954, −9.597790390259362621458842227693, −9.165378066107544715734586675306, −7.929082517893968191050195100128, −7.12345781754165329060112284166, −5.99291504974358417261356609866, −5.22969189092878489635355085946, −3.67152555126827602739945331809, −2.81190907382411647530473884729, −1.52056718611942454322189858960, 1.52056718611942454322189858960, 2.81190907382411647530473884729, 3.67152555126827602739945331809, 5.22969189092878489635355085946, 5.99291504974358417261356609866, 7.12345781754165329060112284166, 7.929082517893968191050195100128, 9.165378066107544715734586675306, 9.597790390259362621458842227693, 10.07988906742863991427306989954

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