Properties

Label 2-825-1.1-c3-0-26
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  + 2.56·2-s − 3·3-s − 1.43·4-s − 7.68·6-s + 29.1·7-s − 24.1·8-s + 9·9-s − 11·11-s + 4.31·12-s − 39.1·13-s + 74.5·14-s − 50.4·16-s + 46.3·17-s + 23.0·18-s + 43.5·19-s − 87.3·21-s − 28.1·22-s + 91.3·23-s + 72.5·24-s − 100.·26-s − 27·27-s − 41.8·28-s − 185.·29-s − 45.8·31-s + 64.2·32-s + 33·33-s + 118.·34-s + ⋯
L(s)  = 1  + 0.905·2-s − 0.577·3-s − 0.179·4-s − 0.522·6-s + 1.57·7-s − 1.06·8-s + 0.333·9-s − 0.301·11-s + 0.103·12-s − 0.835·13-s + 1.42·14-s − 0.787·16-s + 0.661·17-s + 0.301·18-s + 0.526·19-s − 0.907·21-s − 0.273·22-s + 0.827·23-s + 0.616·24-s − 0.756·26-s − 0.192·27-s − 0.282·28-s − 1.19·29-s − 0.265·31-s + 0.354·32-s + 0.174·33-s + 0.598·34-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

\(L(2)\) \(\approx\) \(2.548621355\)
\(L(\frac12)\) \(\approx\) \(2.548621355\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.56T + 8T^{2} \)
7 \( 1 - 29.1T + 343T^{2} \)
13 \( 1 + 39.1T + 2.19e3T^{2} \)
17 \( 1 - 46.3T + 4.91e3T^{2} \)
19 \( 1 - 43.5T + 6.85e3T^{2} \)
23 \( 1 - 91.3T + 1.21e4T^{2} \)
29 \( 1 + 185.T + 2.43e4T^{2} \)
31 \( 1 + 45.8T + 2.97e4T^{2} \)
37 \( 1 - 177.T + 5.06e4T^{2} \)
41 \( 1 - 232.T + 6.89e4T^{2} \)
43 \( 1 - 64.1T + 7.95e4T^{2} \)
47 \( 1 - 16.9T + 1.03e5T^{2} \)
53 \( 1 + 48.7T + 1.48e5T^{2} \)
59 \( 1 - 200.T + 2.05e5T^{2} \)
61 \( 1 - 521.T + 2.26e5T^{2} \)
67 \( 1 + 338.T + 3.00e5T^{2} \)
71 \( 1 - 318.T + 3.57e5T^{2} \)
73 \( 1 - 1.13e3T + 3.89e5T^{2} \)
79 \( 1 - 824.T + 4.93e5T^{2} \)
83 \( 1 + 731.T + 5.71e5T^{2} \)
89 \( 1 - 902.T + 7.04e5T^{2} \)
97 \( 1 - 1.35e3T + 9.12e5T^{2} \)
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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

−9.879549703216117587010066408154, −9.039928171162925809076004781042, −7.941778801968153104467781814079, −7.27729417442585314005623359407, −5.92239498824741225317861466140, −5.15752270681188957744070231407, −4.75221179719072633600753777248, −3.65786041916406184672783970382, −2.28132211055306567908884272537, −0.816261344864138847409443634594, 0.816261344864138847409443634594, 2.28132211055306567908884272537, 3.65786041916406184672783970382, 4.75221179719072633600753777248, 5.15752270681188957744070231407, 5.92239498824741225317861466140, 7.27729417442585314005623359407, 7.941778801968153104467781814079, 9.039928171162925809076004781042, 9.879549703216117587010066408154

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