Properties

Label 2-736-1.1-c3-0-37
Degree $2$
Conductor $736$
Sign $-1$
Analytic cond. $43.4254$
Root an. cond. $6.58979$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.55·3-s − 15.7·5-s + 25.1·7-s − 20.4·9-s − 15.8·11-s + 15.4·13-s + 40.2·15-s + 56.7·17-s + 107.·19-s − 64.4·21-s + 23·23-s + 122.·25-s + 121.·27-s − 267.·29-s − 35.0·31-s + 40.5·33-s − 396.·35-s + 84.9·37-s − 39.6·39-s + 296.·41-s − 353.·43-s + 322.·45-s + 86.6·47-s + 291.·49-s − 145.·51-s − 126.·53-s + 249.·55-s + ⋯
L(s)  = 1  − 0.492·3-s − 1.40·5-s + 1.35·7-s − 0.757·9-s − 0.434·11-s + 0.330·13-s + 0.693·15-s + 0.810·17-s + 1.29·19-s − 0.669·21-s + 0.208·23-s + 0.982·25-s + 0.865·27-s − 1.71·29-s − 0.203·31-s + 0.213·33-s − 1.91·35-s + 0.377·37-s − 0.162·39-s + 1.13·41-s − 1.25·43-s + 1.06·45-s + 0.268·47-s + 0.849·49-s − 0.398·51-s − 0.328·53-s + 0.611·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $-1$
Analytic conductor: \(43.4254\)
Root analytic conductor: \(6.58979\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 736,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 - 23T \)
good3 \( 1 + 2.55T + 27T^{2} \)
5 \( 1 + 15.7T + 125T^{2} \)
7 \( 1 - 25.1T + 343T^{2} \)
11 \( 1 + 15.8T + 1.33e3T^{2} \)
13 \( 1 - 15.4T + 2.19e3T^{2} \)
17 \( 1 - 56.7T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
29 \( 1 + 267.T + 2.43e4T^{2} \)
31 \( 1 + 35.0T + 2.97e4T^{2} \)
37 \( 1 - 84.9T + 5.06e4T^{2} \)
41 \( 1 - 296.T + 6.89e4T^{2} \)
43 \( 1 + 353.T + 7.95e4T^{2} \)
47 \( 1 - 86.6T + 1.03e5T^{2} \)
53 \( 1 + 126.T + 1.48e5T^{2} \)
59 \( 1 + 853.T + 2.05e5T^{2} \)
61 \( 1 - 647.T + 2.26e5T^{2} \)
67 \( 1 + 603.T + 3.00e5T^{2} \)
71 \( 1 - 467.T + 3.57e5T^{2} \)
73 \( 1 - 301.T + 3.89e5T^{2} \)
79 \( 1 + 766.T + 4.93e5T^{2} \)
83 \( 1 + 660.T + 5.71e5T^{2} \)
89 \( 1 + 1.10e3T + 7.04e5T^{2} \)
97 \( 1 + 1.49e3T + 9.12e5T^{2} \)
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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

−9.518855451499853210226008701921, −8.385947764749532679738460661665, −7.86559883935223023242807760666, −7.23068401466301959060343399269, −5.70170735139579081995071950175, −5.10301890489386640698913793664, −4.04265689658861867933350153999, −3.00677234958593858023656198310, −1.28898606549943203557104086560, 0, 1.28898606549943203557104086560, 3.00677234958593858023656198310, 4.04265689658861867933350153999, 5.10301890489386640698913793664, 5.70170735139579081995071950175, 7.23068401466301959060343399269, 7.86559883935223023242807760666, 8.385947764749532679738460661665, 9.518855451499853210226008701921

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