Properties

Label 2-825-1.1-c5-0-5
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  + 6.71·2-s − 9·3-s + 13.0·4-s − 60.4·6-s − 177.·7-s − 127.·8-s + 81·9-s − 121·11-s − 117.·12-s − 783.·13-s − 1.18e3·14-s − 1.27e3·16-s − 1.33e3·17-s + 543.·18-s − 915.·19-s + 1.59e3·21-s − 812.·22-s − 1.68e3·23-s + 1.14e3·24-s − 5.25e3·26-s − 729·27-s − 2.31e3·28-s + 1.72e3·29-s + 7.74e3·31-s − 4.46e3·32-s + 1.08e3·33-s − 8.96e3·34-s + ⋯
L(s)  = 1  + 1.18·2-s − 0.577·3-s + 0.407·4-s − 0.685·6-s − 1.36·7-s − 0.702·8-s + 0.333·9-s − 0.301·11-s − 0.235·12-s − 1.28·13-s − 1.62·14-s − 1.24·16-s − 1.12·17-s + 0.395·18-s − 0.581·19-s + 0.789·21-s − 0.357·22-s − 0.664·23-s + 0.405·24-s − 1.52·26-s − 0.192·27-s − 0.557·28-s + 0.379·29-s + 1.44·31-s − 0.770·32-s + 0.174·33-s − 1.32·34-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

\(L(3)\) \(\approx\) \(0.6172758815\)
\(L(\frac12)\) \(\approx\) \(0.6172758815\)
\(L(\frac{7}{2})\) 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 \( 1 - 6.71T + 32T^{2} \)
7 \( 1 + 177.T + 1.68e4T^{2} \)
13 \( 1 + 783.T + 3.71e5T^{2} \)
17 \( 1 + 1.33e3T + 1.41e6T^{2} \)
19 \( 1 + 915.T + 2.47e6T^{2} \)
23 \( 1 + 1.68e3T + 6.43e6T^{2} \)
29 \( 1 - 1.72e3T + 2.05e7T^{2} \)
31 \( 1 - 7.74e3T + 2.86e7T^{2} \)
37 \( 1 + 9.01e3T + 6.93e7T^{2} \)
41 \( 1 - 1.38e4T + 1.15e8T^{2} \)
43 \( 1 - 1.18e4T + 1.47e8T^{2} \)
47 \( 1 + 2.05e4T + 2.29e8T^{2} \)
53 \( 1 + 3.48e3T + 4.18e8T^{2} \)
59 \( 1 + 2.22e4T + 7.14e8T^{2} \)
61 \( 1 + 4.08e4T + 8.44e8T^{2} \)
67 \( 1 - 2.69e3T + 1.35e9T^{2} \)
71 \( 1 - 1.32e4T + 1.80e9T^{2} \)
73 \( 1 + 3.72e4T + 2.07e9T^{2} \)
79 \( 1 + 4.62e4T + 3.07e9T^{2} \)
83 \( 1 - 401.T + 3.93e9T^{2} \)
89 \( 1 + 3.34e4T + 5.58e9T^{2} \)
97 \( 1 - 1.68e5T + 8.58e9T^{2} \)
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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

−9.646182882098672942944262050785, −8.765009476335041583857157312861, −7.39422196696488301497547173171, −6.42809993256306995606756918299, −6.05792419175022643144537938338, −4.86372728685317021716955222237, −4.31466053311774513227572846435, −3.14449623359567758439067793339, −2.32436192962060808319426831318, −0.28261528811720607172130212163, 0.28261528811720607172130212163, 2.32436192962060808319426831318, 3.14449623359567758439067793339, 4.31466053311774513227572846435, 4.86372728685317021716955222237, 6.05792419175022643144537938338, 6.42809993256306995606756918299, 7.39422196696488301497547173171, 8.765009476335041583857157312861, 9.646182882098672942944262050785

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