Properties

Label 2-825-1.1-c5-0-114
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  + 11.0·2-s − 9·3-s + 89.3·4-s − 99.1·6-s + 62.7·7-s + 631.·8-s + 81·9-s + 121·11-s − 803.·12-s + 986.·13-s + 691.·14-s + 4.09e3·16-s + 577.·17-s + 892.·18-s − 234.·19-s − 564.·21-s + 1.33e3·22-s − 2.02e3·23-s − 5.68e3·24-s + 1.08e4·26-s − 729·27-s + 5.60e3·28-s − 6.87e3·29-s − 689.·31-s + 2.49e4·32-s − 1.08e3·33-s + 6.36e3·34-s + ⋯
L(s)  = 1  + 1.94·2-s − 0.577·3-s + 2.79·4-s − 1.12·6-s + 0.484·7-s + 3.48·8-s + 0.333·9-s + 0.301·11-s − 1.61·12-s + 1.61·13-s + 0.942·14-s + 4.00·16-s + 0.485·17-s + 0.649·18-s − 0.149·19-s − 0.279·21-s + 0.587·22-s − 0.797·23-s − 2.01·24-s + 3.15·26-s − 0.192·27-s + 1.35·28-s − 1.51·29-s − 0.128·31-s + 4.30·32-s − 0.174·33-s + 0.944·34-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

\(L(3)\) \(\approx\) \(9.249460351\)
\(L(\frac12)\) \(\approx\) \(9.249460351\)
\(L(\frac{7}{2})\) 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 \( 1 - 11.0T + 32T^{2} \)
7 \( 1 - 62.7T + 1.68e4T^{2} \)
13 \( 1 - 986.T + 3.71e5T^{2} \)
17 \( 1 - 577.T + 1.41e6T^{2} \)
19 \( 1 + 234.T + 2.47e6T^{2} \)
23 \( 1 + 2.02e3T + 6.43e6T^{2} \)
29 \( 1 + 6.87e3T + 2.05e7T^{2} \)
31 \( 1 + 689.T + 2.86e7T^{2} \)
37 \( 1 - 1.24e4T + 6.93e7T^{2} \)
41 \( 1 + 5.37e3T + 1.15e8T^{2} \)
43 \( 1 - 1.37e4T + 1.47e8T^{2} \)
47 \( 1 + 2.26e4T + 2.29e8T^{2} \)
53 \( 1 + 5.32e3T + 4.18e8T^{2} \)
59 \( 1 - 2.37e4T + 7.14e8T^{2} \)
61 \( 1 - 4.59e4T + 8.44e8T^{2} \)
67 \( 1 - 5.05e4T + 1.35e9T^{2} \)
71 \( 1 + 2.93e4T + 1.80e9T^{2} \)
73 \( 1 + 4.24e4T + 2.07e9T^{2} \)
79 \( 1 - 6.38e4T + 3.07e9T^{2} \)
83 \( 1 + 5.79e4T + 3.93e9T^{2} \)
89 \( 1 - 7.50e4T + 5.58e9T^{2} \)
97 \( 1 + 1.08e4T + 8.58e9T^{2} \)
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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

−9.837773427783585558762011439179, −8.268590148074871396265844359895, −7.39712978176897804802104529219, −6.35125006272368892452558838225, −5.88970776680153317849817907732, −5.05468876037070679366553620998, −4.06351561702145804367409535231, −3.50196887103602153843138038265, −2.08597934501081887978949251948, −1.17104405047908976279181639084, 1.17104405047908976279181639084, 2.08597934501081887978949251948, 3.50196887103602153843138038265, 4.06351561702145804367409535231, 5.05468876037070679366553620998, 5.88970776680153317849817907732, 6.35125006272368892452558838225, 7.39712978176897804802104529219, 8.268590148074871396265844359895, 9.837773427783585558762011439179

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