Properties

Label 2-7500-1.1-c1-0-9
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 − 3.54·7-s + 9-s − 2.20·11-s − 7.17·13-s + 6.36·17-s − 2.31·19-s − 3.54·21-s + 2.17·23-s + 27-s + 0.847·29-s − 4.30·31-s − 2.20·33-s − 7.22·37-s − 7.17·39-s − 1.34·41-s + 8.18·43-s + 6.05·47-s + 5.58·49-s + 6.36·51-s + 11.9·53-s − 2.31·57-s − 12.4·59-s − 6.92·61-s − 3.54·63-s + 4.73·67-s + 2.17·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·7-s + 0.333·9-s − 0.665·11-s − 1.99·13-s + 1.54·17-s − 0.530·19-s − 0.774·21-s + 0.453·23-s + 0.192·27-s + 0.157·29-s − 0.772·31-s − 0.384·33-s − 1.18·37-s − 1.14·39-s − 0.209·41-s + 1.24·43-s + 0.883·47-s + 0.797·49-s + 0.891·51-s + 1.64·53-s − 0.306·57-s − 1.62·59-s − 0.886·61-s − 0.446·63-s + 0.578·67-s + 0.261·69-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

\(L(1)\) \(\approx\) \(1.376488613\)
\(L(\frac12)\) \(\approx\) \(1.376488613\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 + 2.20T + 11T^{2} \)
13 \( 1 + 7.17T + 13T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
19 \( 1 + 2.31T + 19T^{2} \)
23 \( 1 - 2.17T + 23T^{2} \)
29 \( 1 - 0.847T + 29T^{2} \)
31 \( 1 + 4.30T + 31T^{2} \)
37 \( 1 + 7.22T + 37T^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 - 8.18T + 43T^{2} \)
47 \( 1 - 6.05T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 - 4.73T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 1.08T + 73T^{2} \)
79 \( 1 + 5.84T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 7.02T + 89T^{2} \)
97 \( 1 + 18.1T + 97T^{2} \)
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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

−7.67759083378433894797728806128, −7.33160012905300564335476781207, −6.68898379065394421887708229284, −5.68886416301788425498848950804, −5.17160158440300194831045736917, −4.22388139198733691553844006660, −3.31760596015894463646989711883, −2.81397985564879230601524384515, −2.04613697108456957996472743505, −0.53857581947606624304283391254, 0.53857581947606624304283391254, 2.04613697108456957996472743505, 2.81397985564879230601524384515, 3.31760596015894463646989711883, 4.22388139198733691553844006660, 5.17160158440300194831045736917, 5.68886416301788425498848950804, 6.68898379065394421887708229284, 7.33160012905300564335476781207, 7.67759083378433894797728806128

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