Properties

Label 2-7500-1.1-c1-0-36
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 + 4.78·7-s + 9-s + 1.95·11-s − 1.07·13-s + 3.80·17-s + 4.25·19-s − 4.78·21-s + 5.86·23-s − 27-s + 10.5·29-s − 4.31·31-s − 1.95·33-s − 5.65·37-s + 1.07·39-s + 0.858·41-s + 10.8·43-s − 3.00·47-s + 15.8·49-s − 3.80·51-s − 4.22·53-s − 4.25·57-s − 4.77·59-s + 3.63·61-s + 4.78·63-s − 6.93·67-s − 5.86·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.80·7-s + 0.333·9-s + 0.589·11-s − 0.299·13-s + 0.923·17-s + 0.975·19-s − 1.04·21-s + 1.22·23-s − 0.192·27-s + 1.95·29-s − 0.775·31-s − 0.340·33-s − 0.930·37-s + 0.172·39-s + 0.134·41-s + 1.65·43-s − 0.437·47-s + 2.26·49-s − 0.532·51-s − 0.580·53-s − 0.563·57-s − 0.621·59-s + 0.465·61-s + 0.602·63-s − 0.846·67-s − 0.705·69-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

\(L(1)\) \(\approx\) \(2.645840216\)
\(L(\frac12)\) \(\approx\) \(2.645840216\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.78T + 7T^{2} \)
11 \( 1 - 1.95T + 11T^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 - 4.25T + 19T^{2} \)
23 \( 1 - 5.86T + 23T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 0.858T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 + 4.22T + 53T^{2} \)
59 \( 1 + 4.77T + 59T^{2} \)
61 \( 1 - 3.63T + 61T^{2} \)
67 \( 1 + 6.93T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 16.9T + 73T^{2} \)
79 \( 1 + 6.62T + 79T^{2} \)
83 \( 1 - 2.58T + 83T^{2} \)
89 \( 1 + 0.832T + 89T^{2} \)
97 \( 1 - 9.11T + 97T^{2} \)
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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

−7.66419003456306561186708961715, −7.40082359819348899121790620265, −6.49714133497631937597515694750, −5.62336920935959765305586611991, −5.00831141876622118018709305688, −4.62542324069847480491221056547, −3.63902355173159990801115560220, −2.63742428438704496048532887571, −1.46855612731239195679941252030, −0.983696698699724962757799453656, 0.983696698699724962757799453656, 1.46855612731239195679941252030, 2.63742428438704496048532887571, 3.63902355173159990801115560220, 4.62542324069847480491221056547, 5.00831141876622118018709305688, 5.62336920935959765305586611991, 6.49714133497631937597515694750, 7.40082359819348899121790620265, 7.66419003456306561186708961715

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