Properties

Label 2-825-1.1-c5-0-50
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  − 7.81·2-s + 9·3-s + 29.0·4-s − 70.3·6-s + 11.0·7-s + 22.8·8-s + 81·9-s + 121·11-s + 261.·12-s + 248.·13-s − 86.2·14-s − 1.10e3·16-s − 669.·17-s − 633.·18-s + 2.32e3·19-s + 99.2·21-s − 945.·22-s + 867.·23-s + 205.·24-s − 1.94e3·26-s + 729·27-s + 320.·28-s + 2.64e3·29-s − 3.38e3·31-s + 7.93e3·32-s + 1.08e3·33-s + 5.23e3·34-s + ⋯
L(s)  = 1  − 1.38·2-s + 0.577·3-s + 0.908·4-s − 0.797·6-s + 0.0851·7-s + 0.126·8-s + 0.333·9-s + 0.301·11-s + 0.524·12-s + 0.408·13-s − 0.117·14-s − 1.08·16-s − 0.561·17-s − 0.460·18-s + 1.47·19-s + 0.0491·21-s − 0.416·22-s + 0.342·23-s + 0.0729·24-s − 0.564·26-s + 0.192·27-s + 0.0773·28-s + 0.583·29-s − 0.631·31-s + 1.36·32-s + 0.174·33-s + 0.775·34-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

\(L(3)\) \(\approx\) \(1.467418283\)
\(L(\frac12)\) \(\approx\) \(1.467418283\)
\(L(\frac{7}{2})\) 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 \( 1 + 7.81T + 32T^{2} \)
7 \( 1 - 11.0T + 1.68e4T^{2} \)
13 \( 1 - 248.T + 3.71e5T^{2} \)
17 \( 1 + 669.T + 1.41e6T^{2} \)
19 \( 1 - 2.32e3T + 2.47e6T^{2} \)
23 \( 1 - 867.T + 6.43e6T^{2} \)
29 \( 1 - 2.64e3T + 2.05e7T^{2} \)
31 \( 1 + 3.38e3T + 2.86e7T^{2} \)
37 \( 1 + 434.T + 6.93e7T^{2} \)
41 \( 1 - 6.11e3T + 1.15e8T^{2} \)
43 \( 1 - 6.68e3T + 1.47e8T^{2} \)
47 \( 1 - 2.01e4T + 2.29e8T^{2} \)
53 \( 1 - 3.33e3T + 4.18e8T^{2} \)
59 \( 1 + 3.56e4T + 7.14e8T^{2} \)
61 \( 1 - 1.67e4T + 8.44e8T^{2} \)
67 \( 1 - 1.50e4T + 1.35e9T^{2} \)
71 \( 1 + 2.10e4T + 1.80e9T^{2} \)
73 \( 1 - 5.36e3T + 2.07e9T^{2} \)
79 \( 1 - 1.17e4T + 3.07e9T^{2} \)
83 \( 1 - 5.19e4T + 3.93e9T^{2} \)
89 \( 1 - 8.96e4T + 5.58e9T^{2} \)
97 \( 1 - 1.94e4T + 8.58e9T^{2} \)
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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

−9.223433940686696835813409588752, −8.877399333135702794896121700852, −7.87122254018309794140804354326, −7.33065972556496908697637154149, −6.35982099749689406094836310953, −5.01845226647345704192770257505, −3.88430447950666962512712050400, −2.69037287059386418678874860062, −1.56004984908802056305050724872, −0.70634298584810316347685686403, 0.70634298584810316347685686403, 1.56004984908802056305050724872, 2.69037287059386418678874860062, 3.88430447950666962512712050400, 5.01845226647345704192770257505, 6.35982099749689406094836310953, 7.33065972556496908697637154149, 7.87122254018309794140804354326, 8.877399333135702794896121700852, 9.223433940686696835813409588752

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