Properties

Label 2-7350-1.1-c1-0-43
Degree $2$
Conductor $7350$
Sign $1$
Analytic cond. $58.6900$
Root an. cond. $7.66094$
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  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 4.78·11-s + 12-s − 3.17·13-s + 16-s + 5.22·17-s + 18-s − 3.17·19-s − 4.78·22-s + 7.17·23-s + 24-s − 3.17·26-s + 27-s + 2.38·29-s + 4.17·31-s + 32-s − 4.78·33-s + 5.22·34-s + 36-s + 7.17·37-s − 3.17·38-s − 3.17·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.44·11-s + 0.288·12-s − 0.879·13-s + 0.250·16-s + 1.26·17-s + 0.235·18-s − 0.727·19-s − 1.02·22-s + 1.49·23-s + 0.204·24-s − 0.621·26-s + 0.192·27-s + 0.443·29-s + 0.749·31-s + 0.176·32-s − 0.832·33-s + 0.896·34-s + 0.166·36-s + 1.17·37-s − 0.514·38-s − 0.507·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(58.6900\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.846307924\)
\(L(\frac12)\) \(\approx\) \(3.846307924\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 4.78T + 11T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 + 3.17T + 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 - 2.38T + 29T^{2} \)
31 \( 1 - 4.17T + 31T^{2} \)
37 \( 1 - 7.17T + 37T^{2} \)
41 \( 1 + 2.05T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + 8.11T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 7.11T + 61T^{2} \)
67 \( 1 - 9.56T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 3.39T + 83T^{2} \)
89 \( 1 - 9.56T + 89T^{2} \)
97 \( 1 + 1.27T + 97T^{2} \)
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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

−7.88846707595273381235453955846, −7.25968511218519641209058110691, −6.54047211443070837462791610159, −5.62300045209755651968906724899, −5.00143518303588654245870807887, −4.46540755473213992444800790998, −3.38644200586490111580477367807, −2.78753895184083784276427500983, −2.19477984177765303675755027586, −0.854488245661162018636994088215, 0.854488245661162018636994088215, 2.19477984177765303675755027586, 2.78753895184083784276427500983, 3.38644200586490111580477367807, 4.46540755473213992444800790998, 5.00143518303588654245870807887, 5.62300045209755651968906724899, 6.54047211443070837462791610159, 7.25968511218519641209058110691, 7.88846707595273381235453955846

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