Properties

Label 2-825-1.1-c3-0-66
Degree $2$
Conductor $825$
Sign $-1$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.42·2-s + 3·3-s + 21.4·4-s − 16.2·6-s + 7.69·7-s − 72.8·8-s + 9·9-s − 11·11-s + 64.2·12-s − 24.8·13-s − 41.7·14-s + 223.·16-s + 15.9·17-s − 48.8·18-s + 15.1·19-s + 23.0·21-s + 59.6·22-s − 17.7·23-s − 218.·24-s + 134.·26-s + 27·27-s + 164.·28-s − 128.·29-s + 219.·31-s − 630.·32-s − 33·33-s − 86.4·34-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.577·3-s + 2.67·4-s − 1.10·6-s + 0.415·7-s − 3.21·8-s + 0.333·9-s − 0.301·11-s + 1.54·12-s − 0.530·13-s − 0.797·14-s + 3.49·16-s + 0.227·17-s − 0.639·18-s + 0.182·19-s + 0.239·21-s + 0.578·22-s − 0.160·23-s − 1.85·24-s + 1.01·26-s + 0.192·27-s + 1.11·28-s − 0.823·29-s + 1.27·31-s − 3.48·32-s − 0.174·33-s − 0.436·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 + 5.42T + 8T^{2} \)
7 \( 1 - 7.69T + 343T^{2} \)
13 \( 1 + 24.8T + 2.19e3T^{2} \)
17 \( 1 - 15.9T + 4.91e3T^{2} \)
19 \( 1 - 15.1T + 6.85e3T^{2} \)
23 \( 1 + 17.7T + 1.21e4T^{2} \)
29 \( 1 + 128.T + 2.43e4T^{2} \)
31 \( 1 - 219.T + 2.97e4T^{2} \)
37 \( 1 + 92.0T + 5.06e4T^{2} \)
41 \( 1 + 459.T + 6.89e4T^{2} \)
43 \( 1 + 64.9T + 7.95e4T^{2} \)
47 \( 1 + 497.T + 1.03e5T^{2} \)
53 \( 1 - 526.T + 1.48e5T^{2} \)
59 \( 1 + 578.T + 2.05e5T^{2} \)
61 \( 1 + 221.T + 2.26e5T^{2} \)
67 \( 1 - 860.T + 3.00e5T^{2} \)
71 \( 1 - 580.T + 3.57e5T^{2} \)
73 \( 1 + 510.T + 3.89e5T^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 606.T + 5.71e5T^{2} \)
89 \( 1 + 23.4T + 7.04e5T^{2} \)
97 \( 1 + 719.T + 9.12e5T^{2} \)
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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

−9.469347718018433822643077355653, −8.442709499838904485943608224778, −8.047811186628844589726066867079, −7.21961937914487229477665703058, −6.42431338613197450313024015287, −5.09406308198857830745715990997, −3.35637830853736549548861610668, −2.30985732823011597579409948220, −1.37992906861828095508025257151, 0, 1.37992906861828095508025257151, 2.30985732823011597579409948220, 3.35637830853736549548861610668, 5.09406308198857830745715990997, 6.42431338613197450313024015287, 7.21961937914487229477665703058, 8.047811186628844589726066867079, 8.442709499838904485943608224778, 9.469347718018433822643077355653

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