Properties

Label 2-7500-1.1-c1-0-30
Degree $2$
Conductor $7500$
Sign $1$
Analytic cond. $59.8878$
Root an. cond. $7.73872$
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  + 3-s + 3.80·7-s + 9-s + 0.190·11-s + 1.67·13-s − 4.60·17-s − 2.64·19-s + 3.80·21-s − 6.35·23-s + 27-s + 2.52·29-s + 3.74·31-s + 0.190·33-s + 11.8·37-s + 1.67·39-s − 7.18·41-s + 9.22·43-s + 4.54·47-s + 7.51·49-s − 4.60·51-s − 9.43·53-s − 2.64·57-s + 7.14·59-s + 9.53·61-s + 3.80·63-s − 6.05·67-s − 6.35·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.44·7-s + 0.333·9-s + 0.0575·11-s + 0.465·13-s − 1.11·17-s − 0.605·19-s + 0.831·21-s − 1.32·23-s + 0.192·27-s + 0.467·29-s + 0.673·31-s + 0.0332·33-s + 1.95·37-s + 0.268·39-s − 1.12·41-s + 1.40·43-s + 0.663·47-s + 1.07·49-s − 0.644·51-s − 1.29·53-s − 0.349·57-s + 0.929·59-s + 1.22·61-s + 0.480·63-s − 0.739·67-s − 0.765·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7500\)    =    \(2^{2} \cdot 3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(59.8878\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7500,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.149548976\)
\(L(\frac12)\) \(\approx\) \(3.149548976\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 3.80T + 7T^{2} \)
11 \( 1 - 0.190T + 11T^{2} \)
13 \( 1 - 1.67T + 13T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 + 2.64T + 19T^{2} \)
23 \( 1 + 6.35T + 23T^{2} \)
29 \( 1 - 2.52T + 29T^{2} \)
31 \( 1 - 3.74T + 31T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 + 7.18T + 41T^{2} \)
43 \( 1 - 9.22T + 43T^{2} \)
47 \( 1 - 4.54T + 47T^{2} \)
53 \( 1 + 9.43T + 53T^{2} \)
59 \( 1 - 7.14T + 59T^{2} \)
61 \( 1 - 9.53T + 61T^{2} \)
67 \( 1 + 6.05T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 5.21T + 73T^{2} \)
79 \( 1 + 3.11T + 79T^{2} \)
83 \( 1 + 4.67T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 + 6.69T + 97T^{2} \)
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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

−8.059542070312229571378668852030, −7.38813675519615142209666422013, −6.46335109025023758268905410520, −5.90172778613048595108352752369, −4.79072109744276986915394408131, −4.40332458774460414493069438280, −3.66764824203792428844651190256, −2.42354834683020272435061069627, −1.99094026514467808026147456972, −0.888997884612696181141123877305, 0.888997884612696181141123877305, 1.99094026514467808026147456972, 2.42354834683020272435061069627, 3.66764824203792428844651190256, 4.40332458774460414493069438280, 4.79072109744276986915394408131, 5.90172778613048595108352752369, 6.46335109025023758268905410520, 7.38813675519615142209666422013, 8.059542070312229571378668852030

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