Properties

Label 2-7448-1.1-c1-0-74
Degree $2$
Conductor $7448$
Sign $1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s + 9-s − 3·11-s + 4·13-s + 2·15-s + 2·17-s + 19-s − 7·23-s − 4·25-s − 4·27-s + 2·29-s + 6·31-s − 6·33-s − 10·37-s + 8·39-s + 8·41-s + 7·43-s + 45-s + 9·47-s + 4·51-s + 6·53-s − 3·55-s + 2·57-s + 14·59-s + 5·61-s + 4·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.516·15-s + 0.485·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s − 0.769·27-s + 0.371·29-s + 1.07·31-s − 1.04·33-s − 1.64·37-s + 1.28·39-s + 1.24·41-s + 1.06·43-s + 0.149·45-s + 1.31·47-s + 0.560·51-s + 0.824·53-s − 0.404·55-s + 0.264·57-s + 1.82·59-s + 0.640·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7448\)    =    \(2^{3} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(59.4725\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.374993715\)
\(L(\frac12)\) \(\approx\) \(3.374993715\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−8.052567334422649857826091515646, −7.46712766478967909353633740637, −6.47011709914738638224082363647, −5.76752663525708997195007098229, −5.22313585005548073268036613318, −3.92559668470188645679496190708, −3.64770193165967076405604715660, −2.48597725176336781972653187797, −2.17264391045067968392885390816, −0.870181391233899341603195972834, 0.870181391233899341603195972834, 2.17264391045067968392885390816, 2.48597725176336781972653187797, 3.64770193165967076405604715660, 3.92559668470188645679496190708, 5.22313585005548073268036613318, 5.76752663525708997195007098229, 6.47011709914738638224082363647, 7.46712766478967909353633740637, 8.052567334422649857826091515646

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