Properties

Label 2-7448-1.1-c1-0-5
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  + 0.488·3-s − 3.51·5-s − 2.76·9-s − 2.25·11-s − 2.21·13-s − 1.71·15-s − 0.716·17-s − 19-s − 7.53·23-s + 7.35·25-s − 2.81·27-s − 7.66·29-s + 0.473·31-s − 1.10·33-s − 0.268·37-s − 1.08·39-s − 11.9·41-s − 6.26·43-s + 9.70·45-s + 7.10·47-s − 0.350·51-s − 2.16·53-s + 7.93·55-s − 0.488·57-s − 0.0795·59-s + 15.1·61-s + 7.77·65-s + ⋯
L(s)  = 1  + 0.281·3-s − 1.57·5-s − 0.920·9-s − 0.680·11-s − 0.613·13-s − 0.443·15-s − 0.173·17-s − 0.229·19-s − 1.57·23-s + 1.47·25-s − 0.541·27-s − 1.42·29-s + 0.0851·31-s − 0.191·33-s − 0.0440·37-s − 0.173·39-s − 1.85·41-s − 0.954·43-s + 1.44·45-s + 1.03·47-s − 0.0490·51-s − 0.297·53-s + 1.07·55-s − 0.0646·57-s − 0.0103·59-s + 1.94·61-s + 0.964·65-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: $\chi_{7448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2482963309\)
\(L(\frac12)\) \(\approx\) \(0.2482963309\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.488T + 3T^{2} \)
5 \( 1 + 3.51T + 5T^{2} \)
11 \( 1 + 2.25T + 11T^{2} \)
13 \( 1 + 2.21T + 13T^{2} \)
17 \( 1 + 0.716T + 17T^{2} \)
23 \( 1 + 7.53T + 23T^{2} \)
29 \( 1 + 7.66T + 29T^{2} \)
31 \( 1 - 0.473T + 31T^{2} \)
37 \( 1 + 0.268T + 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + 6.26T + 43T^{2} \)
47 \( 1 - 7.10T + 47T^{2} \)
53 \( 1 + 2.16T + 53T^{2} \)
59 \( 1 + 0.0795T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 - 7.97T + 67T^{2} \)
71 \( 1 - 0.764T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 3.68T + 83T^{2} \)
89 \( 1 - 2.22T + 89T^{2} \)
97 \( 1 - 7.09T + 97T^{2} \)
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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

−7.987137959212935168004729999263, −7.38559522958304915246270365340, −6.68734009973001197959322087771, −5.66821846984933075870213199835, −5.07798495269410581324018695092, −4.12334150719355958617591261930, −3.62627828824985047544388995946, −2.80972643845985062692944197964, −1.95405888892133897879923615743, −0.22944574605280368112405544904, 0.22944574605280368112405544904, 1.95405888892133897879923615743, 2.80972643845985062692944197964, 3.62627828824985047544388995946, 4.12334150719355958617591261930, 5.07798495269410581324018695092, 5.66821846984933075870213199835, 6.68734009973001197959322087771, 7.38559522958304915246270365340, 7.987137959212935168004729999263

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