Properties

Label 2-7448-1.1-c1-0-121
Degree $2$
Conductor $7448$
Sign $1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
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  + 1.82·3-s + 3.85·5-s + 0.313·9-s + 4.19·11-s + 3.06·13-s + 7.01·15-s − 3.97·17-s + 19-s + 4.41·23-s + 9.86·25-s − 4.89·27-s − 1.30·29-s − 2.76·31-s + 7.62·33-s − 6.65·37-s + 5.57·39-s + 7.13·41-s + 0.941·43-s + 1.20·45-s − 3.82·47-s − 7.24·51-s − 4.75·53-s + 16.1·55-s + 1.82·57-s + 14.2·59-s + 12.3·61-s + 11.8·65-s + ⋯
L(s)  = 1  + 1.05·3-s + 1.72·5-s + 0.104·9-s + 1.26·11-s + 0.849·13-s + 1.81·15-s − 0.965·17-s + 0.229·19-s + 0.920·23-s + 1.97·25-s − 0.941·27-s − 0.242·29-s − 0.495·31-s + 1.32·33-s − 1.09·37-s + 0.892·39-s + 1.11·41-s + 0.143·43-s + 0.180·45-s − 0.558·47-s − 1.01·51-s − 0.653·53-s + 2.17·55-s + 0.241·57-s + 1.85·59-s + 1.57·61-s + 1.46·65-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.861029812\)
\(L(\frac12)\) \(\approx\) \(4.861029812\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.82T + 3T^{2} \)
5 \( 1 - 3.85T + 5T^{2} \)
11 \( 1 - 4.19T + 11T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 + 3.97T + 17T^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 + 1.30T + 29T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 + 6.65T + 37T^{2} \)
41 \( 1 - 7.13T + 41T^{2} \)
43 \( 1 - 0.941T + 43T^{2} \)
47 \( 1 + 3.82T + 47T^{2} \)
53 \( 1 + 4.75T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 1.23T + 67T^{2} \)
71 \( 1 - 3.34T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 4.37T + 79T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 - 2.18T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−8.178464662708159604128204517060, −6.90379711861500615695964344581, −6.68696222218702650444926869433, −5.78345182729168492508314701772, −5.26438943951919076982765585933, −4.13523744435984718016003045239, −3.44579904665393142515010379159, −2.55919017812383297123796288074, −1.91761399744038229316703088925, −1.15395910216850831570975151773, 1.15395910216850831570975151773, 1.91761399744038229316703088925, 2.55919017812383297123796288074, 3.44579904665393142515010379159, 4.13523744435984718016003045239, 5.26438943951919076982765585933, 5.78345182729168492508314701772, 6.68696222218702650444926869433, 6.90379711861500615695964344581, 8.178464662708159604128204517060

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