Properties

Label 2-825-1.1-c3-0-1
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·2-s − 3·3-s − 5.56·4-s + 4.68·6-s − 16.7·7-s + 21.1·8-s + 9·9-s − 11·11-s + 16.6·12-s − 61.6·13-s + 26.0·14-s + 11.4·16-s − 81.9·17-s − 14.0·18-s − 66.0·19-s + 50.1·21-s + 17.1·22-s + 118.·23-s − 63.5·24-s + 96.3·26-s − 27·27-s + 92.9·28-s + 4.91·29-s − 286.·31-s − 187.·32-s + 33·33-s + 127.·34-s + ⋯
L(s)  = 1  − 0.552·2-s − 0.577·3-s − 0.695·4-s + 0.318·6-s − 0.902·7-s + 0.935·8-s + 0.333·9-s − 0.301·11-s + 0.401·12-s − 1.31·13-s + 0.497·14-s + 0.178·16-s − 1.16·17-s − 0.184·18-s − 0.797·19-s + 0.520·21-s + 0.166·22-s + 1.06·23-s − 0.540·24-s + 0.726·26-s − 0.192·27-s + 0.627·28-s + 0.0314·29-s − 1.66·31-s − 1.03·32-s + 0.174·33-s + 0.645·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1443558802\)
\(L(\frac12)\) \(\approx\) \(0.1443558802\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 + 1.56T + 8T^{2} \)
7 \( 1 + 16.7T + 343T^{2} \)
13 \( 1 + 61.6T + 2.19e3T^{2} \)
17 \( 1 + 81.9T + 4.91e3T^{2} \)
19 \( 1 + 66.0T + 6.85e3T^{2} \)
23 \( 1 - 118.T + 1.21e4T^{2} \)
29 \( 1 - 4.91T + 2.43e4T^{2} \)
31 \( 1 + 286.T + 2.97e4T^{2} \)
37 \( 1 + 271.T + 5.06e4T^{2} \)
41 \( 1 - 117.T + 6.89e4T^{2} \)
43 \( 1 + 364.T + 7.95e4T^{2} \)
47 \( 1 + 465.T + 1.03e5T^{2} \)
53 \( 1 + 600.T + 1.48e5T^{2} \)
59 \( 1 - 647.T + 2.05e5T^{2} \)
61 \( 1 + 190.T + 2.26e5T^{2} \)
67 \( 1 + 1.08e3T + 3.00e5T^{2} \)
71 \( 1 - 811.T + 3.57e5T^{2} \)
73 \( 1 - 1.03e3T + 3.89e5T^{2} \)
79 \( 1 - 573.T + 4.93e5T^{2} \)
83 \( 1 - 510.T + 5.71e5T^{2} \)
89 \( 1 + 748.T + 7.04e5T^{2} \)
97 \( 1 + 96.0T + 9.12e5T^{2} \)
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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

−9.705672268703439525959233031038, −9.206119933152011379234952163757, −8.260371665180794171644346922846, −7.18416121951226744993364548058, −6.56420888180348765246567858192, −5.23950233873534650174201224221, −4.64063382901171336684551352692, −3.40809174806615419809050695564, −1.93292772264029760708719137861, −0.22766649383122167304307133668, 0.22766649383122167304307133668, 1.93292772264029760708719137861, 3.40809174806615419809050695564, 4.64063382901171336684551352692, 5.23950233873534650174201224221, 6.56420888180348765246567858192, 7.18416121951226744993364548058, 8.260371665180794171644346922846, 9.206119933152011379234952163757, 9.705672268703439525959233031038

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