Properties

Label 2-825-1.1-c5-0-151
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.96·2-s − 9·3-s + 48.3·4-s − 80.6·6-s + 191.·7-s + 146.·8-s + 81·9-s − 121·11-s − 434.·12-s − 257.·13-s + 1.71e3·14-s − 235.·16-s − 1.24e3·17-s + 725.·18-s − 2.95e3·19-s − 1.72e3·21-s − 1.08e3·22-s + 374.·23-s − 1.31e3·24-s − 2.30e3·26-s − 729·27-s + 9.25e3·28-s + 4.57e3·29-s + 2.52e3·31-s − 6.79e3·32-s + 1.08e3·33-s − 1.11e4·34-s + ⋯
L(s)  = 1  + 1.58·2-s − 0.577·3-s + 1.50·4-s − 0.914·6-s + 1.47·7-s + 0.807·8-s + 0.333·9-s − 0.301·11-s − 0.871·12-s − 0.422·13-s + 2.33·14-s − 0.230·16-s − 1.04·17-s + 0.528·18-s − 1.87·19-s − 0.852·21-s − 0.477·22-s + 0.147·23-s − 0.466·24-s − 0.669·26-s − 0.192·27-s + 2.22·28-s + 1.00·29-s + 0.471·31-s − 1.17·32-s + 0.174·33-s − 1.65·34-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: \(1\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 8.96T + 32T^{2} \)
7 \( 1 - 191.T + 1.68e4T^{2} \)
13 \( 1 + 257.T + 3.71e5T^{2} \)
17 \( 1 + 1.24e3T + 1.41e6T^{2} \)
19 \( 1 + 2.95e3T + 2.47e6T^{2} \)
23 \( 1 - 374.T + 6.43e6T^{2} \)
29 \( 1 - 4.57e3T + 2.05e7T^{2} \)
31 \( 1 - 2.52e3T + 2.86e7T^{2} \)
37 \( 1 + 1.13e4T + 6.93e7T^{2} \)
41 \( 1 + 7.60e3T + 1.15e8T^{2} \)
43 \( 1 - 853.T + 1.47e8T^{2} \)
47 \( 1 + 1.85e4T + 2.29e8T^{2} \)
53 \( 1 - 1.55e4T + 4.18e8T^{2} \)
59 \( 1 + 1.39e4T + 7.14e8T^{2} \)
61 \( 1 - 2.13e4T + 8.44e8T^{2} \)
67 \( 1 + 5.53e4T + 1.35e9T^{2} \)
71 \( 1 + 7.20e4T + 1.80e9T^{2} \)
73 \( 1 - 5.72e4T + 2.07e9T^{2} \)
79 \( 1 - 7.51e4T + 3.07e9T^{2} \)
83 \( 1 - 1.02e5T + 3.93e9T^{2} \)
89 \( 1 - 1.13e4T + 5.58e9T^{2} \)
97 \( 1 - 2.33e4T + 8.58e9T^{2} \)
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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

−8.874154847641494873857576913374, −8.064576796026207322075858853262, −6.87208703749901579425449967672, −6.28131403184327209375418150235, −5.09545296970282837345984979946, −4.77080142173074292849497027883, −3.97779249968120383590646158520, −2.52534386033468475268684915978, −1.71302000660119052328922160100, 0, 1.71302000660119052328922160100, 2.52534386033468475268684915978, 3.97779249968120383590646158520, 4.77080142173074292849497027883, 5.09545296970282837345984979946, 6.28131403184327209375418150235, 6.87208703749901579425449967672, 8.064576796026207322075858853262, 8.874154847641494873857576913374

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