Properties

Label 2-624-1.1-c3-0-12
Degree $2$
Conductor $624$
Sign $1$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·5-s + 32·7-s + 9·9-s + 68·11-s + 13·13-s + 6·15-s − 14·17-s − 4·19-s − 96·21-s − 72·23-s − 121·25-s − 27·27-s + 102·29-s + 136·31-s − 204·33-s − 64·35-s − 386·37-s − 39·39-s + 250·41-s + 140·43-s − 18·45-s + 296·47-s + 681·49-s + 42·51-s + 526·53-s − 136·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.178·5-s + 1.72·7-s + 1/3·9-s + 1.86·11-s + 0.277·13-s + 0.103·15-s − 0.199·17-s − 0.0482·19-s − 0.997·21-s − 0.652·23-s − 0.967·25-s − 0.192·27-s + 0.653·29-s + 0.787·31-s − 1.07·33-s − 0.309·35-s − 1.71·37-s − 0.160·39-s + 0.952·41-s + 0.496·43-s − 0.0596·45-s + 0.918·47-s + 1.98·49-s + 0.115·51-s + 1.36·53-s − 0.333·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.253356936\)
\(L(\frac12)\) \(\approx\) \(2.253356936\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T \)
13 \( 1 - p T \)
good5 \( 1 + 2 T + p^{3} T^{2} \)
7 \( 1 - 32 T + p^{3} T^{2} \)
11 \( 1 - 68 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 4 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 - 102 T + p^{3} T^{2} \)
31 \( 1 - 136 T + p^{3} T^{2} \)
37 \( 1 + 386 T + p^{3} T^{2} \)
41 \( 1 - 250 T + p^{3} T^{2} \)
43 \( 1 - 140 T + p^{3} T^{2} \)
47 \( 1 - 296 T + p^{3} T^{2} \)
53 \( 1 - 526 T + p^{3} T^{2} \)
59 \( 1 + 332 T + p^{3} T^{2} \)
61 \( 1 + 410 T + p^{3} T^{2} \)
67 \( 1 + 596 T + p^{3} T^{2} \)
71 \( 1 - 880 T + p^{3} T^{2} \)
73 \( 1 - 506 T + p^{3} T^{2} \)
79 \( 1 - 640 T + p^{3} T^{2} \)
83 \( 1 + 1380 T + p^{3} T^{2} \)
89 \( 1 - 1450 T + p^{3} T^{2} \)
97 \( 1 + 446 T + p^{3} T^{2} \)
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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

−10.43584105634642501630546701925, −9.239952523435116294127100265018, −8.458317573968888049386172571596, −7.57017251163248160171193736005, −6.57106881741371067901204111977, −5.65359499739737476567953358731, −4.53432855048527714962421070948, −3.91077623003171798674865183116, −1.93307620765322807646208649545, −1.00328715272185609736644672804, 1.00328715272185609736644672804, 1.93307620765322807646208649545, 3.91077623003171798674865183116, 4.53432855048527714962421070948, 5.65359499739737476567953358731, 6.57106881741371067901204111977, 7.57017251163248160171193736005, 8.458317573968888049386172571596, 9.239952523435116294127100265018, 10.43584105634642501630546701925

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