Properties

Label 2-825-1.1-c5-0-59
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $132.316$
Root an. cond. $11.5028$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.444·2-s + 9·3-s − 31.8·4-s + 4.00·6-s + 205.·7-s − 28.3·8-s + 81·9-s + 121·11-s − 286.·12-s − 893.·13-s + 91.5·14-s + 1.00e3·16-s + 836.·17-s + 36.0·18-s + 816.·19-s + 1.85e3·21-s + 53.8·22-s + 3.12e3·23-s − 255.·24-s − 397.·26-s + 729·27-s − 6.54e3·28-s − 7.30e3·29-s + 152.·31-s + 1.35e3·32-s + 1.08e3·33-s + 371.·34-s + ⋯
L(s)  = 1  + 0.0786·2-s + 0.577·3-s − 0.993·4-s + 0.0453·6-s + 1.58·7-s − 0.156·8-s + 0.333·9-s + 0.301·11-s − 0.573·12-s − 1.46·13-s + 0.124·14-s + 0.981·16-s + 0.701·17-s + 0.0262·18-s + 0.518·19-s + 0.916·21-s + 0.0236·22-s + 1.23·23-s − 0.0904·24-s − 0.115·26-s + 0.192·27-s − 1.57·28-s − 1.61·29-s + 0.0284·31-s + 0.233·32-s + 0.174·33-s + 0.0551·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(132.316\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.847907803\)
\(L(\frac12)\) \(\approx\) \(2.847907803\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
5 \( 1 \)
11 \( 1 - 121T \)
good2 \( 1 - 0.444T + 32T^{2} \)
7 \( 1 - 205.T + 1.68e4T^{2} \)
13 \( 1 + 893.T + 3.71e5T^{2} \)
17 \( 1 - 836.T + 1.41e6T^{2} \)
19 \( 1 - 816.T + 2.47e6T^{2} \)
23 \( 1 - 3.12e3T + 6.43e6T^{2} \)
29 \( 1 + 7.30e3T + 2.05e7T^{2} \)
31 \( 1 - 152.T + 2.86e7T^{2} \)
37 \( 1 - 8.73e3T + 6.93e7T^{2} \)
41 \( 1 + 9.60e3T + 1.15e8T^{2} \)
43 \( 1 - 9.20e3T + 1.47e8T^{2} \)
47 \( 1 - 1.18e4T + 2.29e8T^{2} \)
53 \( 1 - 1.28e4T + 4.18e8T^{2} \)
59 \( 1 + 517.T + 7.14e8T^{2} \)
61 \( 1 + 5.02e4T + 8.44e8T^{2} \)
67 \( 1 - 4.05e3T + 1.35e9T^{2} \)
71 \( 1 - 1.64e4T + 1.80e9T^{2} \)
73 \( 1 + 5.51e4T + 2.07e9T^{2} \)
79 \( 1 - 2.50e4T + 3.07e9T^{2} \)
83 \( 1 + 7.61e4T + 3.93e9T^{2} \)
89 \( 1 - 1.04e5T + 5.58e9T^{2} \)
97 \( 1 - 3.93e4T + 8.58e9T^{2} \)
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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

−9.324609127009521429406683088209, −8.723283384278744615688929217730, −7.67979815676199630384911258267, −7.41292329761360078479397692965, −5.61141552618587265879910390456, −4.91885497350141064605944626303, −4.23149584229061417711772093429, −3.06085858167437547892041310405, −1.81372353844638734716453055318, −0.77528591908715692995858870871, 0.77528591908715692995858870871, 1.81372353844638734716453055318, 3.06085858167437547892041310405, 4.23149584229061417711772093429, 4.91885497350141064605944626303, 5.61141552618587265879910390456, 7.41292329761360078479397692965, 7.67979815676199630384911258267, 8.723283384278744615688929217730, 9.324609127009521429406683088209

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