Properties

Label 2-7800-1.1-c1-0-43
Degree $2$
Conductor $7800$
Sign $1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
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 + 4.96·7-s + 9-s + 0.430·11-s + 13-s − 3·17-s + 3.39·19-s − 4.96·21-s + 5.39·23-s − 27-s − 4.13·29-s + 0.430·31-s − 0.430·33-s + 8.53·37-s − 39-s − 6.53·41-s + 6.53·43-s − 12.1·47-s + 17.6·49-s + 3·51-s + 4.39·53-s − 3.39·57-s + 12.1·59-s + 6.13·61-s + 4.96·63-s − 9.56·67-s − 5.39·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.87·7-s + 0.333·9-s + 0.129·11-s + 0.277·13-s − 0.727·17-s + 0.779·19-s − 1.08·21-s + 1.12·23-s − 0.192·27-s − 0.768·29-s + 0.0773·31-s − 0.0749·33-s + 1.40·37-s − 0.160·39-s − 1.02·41-s + 0.996·43-s − 1.76·47-s + 2.52·49-s + 0.420·51-s + 0.604·53-s − 0.450·57-s + 1.57·59-s + 0.785·61-s + 0.625·63-s − 1.16·67-s − 0.649·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.371597264\)
\(L(\frac12)\) \(\approx\) \(2.371597264\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 - 4.96T + 7T^{2} \)
11 \( 1 - 0.430T + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 3.39T + 19T^{2} \)
23 \( 1 - 5.39T + 23T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 - 0.430T + 31T^{2} \)
37 \( 1 - 8.53T + 37T^{2} \)
41 \( 1 + 6.53T + 41T^{2} \)
43 \( 1 - 6.53T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 4.39T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 6.13T + 61T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 - 9.67T + 71T^{2} \)
73 \( 1 + 9.67T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 7.82T + 83T^{2} \)
89 \( 1 - 2.86T + 89T^{2} \)
97 \( 1 - 5.93T + 97T^{2} \)
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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

−7.83546194131272974913739400765, −7.20838727495613601108673719250, −6.50632224340359983542609703050, −5.56015353867194372565392774434, −5.08261257586526528746151853632, −4.49441171111645072918124767353, −3.71395206035176505803930664235, −2.50657134113328661151702658932, −1.62145352389692138147839197307, −0.857419905429107983260192090025, 0.857419905429107983260192090025, 1.62145352389692138147839197307, 2.50657134113328661151702658932, 3.71395206035176505803930664235, 4.49441171111645072918124767353, 5.08261257586526528746151853632, 5.56015353867194372565392774434, 6.50632224340359983542609703050, 7.20838727495613601108673719250, 7.83546194131272974913739400765

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