Properties

Label 2-825-1.1-c3-0-93
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  + 4.42·2-s + 3·3-s + 11.5·4-s + 13.2·6-s − 31.6·7-s + 15.8·8-s + 9·9-s − 11·11-s + 34.7·12-s − 5.15·13-s − 140.·14-s − 22.6·16-s − 121.·17-s + 39.8·18-s + 34.8·19-s − 95.0·21-s − 48.6·22-s − 116.·23-s + 47.4·24-s − 22.7·26-s + 27·27-s − 366.·28-s − 69.4·29-s + 140.·31-s − 226.·32-s − 33·33-s − 539.·34-s + ⋯
L(s)  = 1  + 1.56·2-s + 0.577·3-s + 1.44·4-s + 0.903·6-s − 1.71·7-s + 0.699·8-s + 0.333·9-s − 0.301·11-s + 0.835·12-s − 0.109·13-s − 2.67·14-s − 0.353·16-s − 1.73·17-s + 0.521·18-s + 0.420·19-s − 0.988·21-s − 0.471·22-s − 1.05·23-s + 0.403·24-s − 0.171·26-s + 0.192·27-s − 2.47·28-s − 0.444·29-s + 0.814·31-s − 1.25·32-s − 0.174·33-s − 2.72·34-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 \( 1 - 4.42T + 8T^{2} \)
7 \( 1 + 31.6T + 343T^{2} \)
13 \( 1 + 5.15T + 2.19e3T^{2} \)
17 \( 1 + 121.T + 4.91e3T^{2} \)
19 \( 1 - 34.8T + 6.85e3T^{2} \)
23 \( 1 + 116.T + 1.21e4T^{2} \)
29 \( 1 + 69.4T + 2.43e4T^{2} \)
31 \( 1 - 140.T + 2.97e4T^{2} \)
37 \( 1 - 420.T + 5.06e4T^{2} \)
41 \( 1 + 322.T + 6.89e4T^{2} \)
43 \( 1 + 321.T + 7.95e4T^{2} \)
47 \( 1 - 231.T + 1.03e5T^{2} \)
53 \( 1 + 4.91T + 1.48e5T^{2} \)
59 \( 1 - 406.T + 2.05e5T^{2} \)
61 \( 1 + 556.T + 2.26e5T^{2} \)
67 \( 1 + 84.7T + 3.00e5T^{2} \)
71 \( 1 - 49.0T + 3.57e5T^{2} \)
73 \( 1 + 785.T + 3.89e5T^{2} \)
79 \( 1 + 383.T + 4.93e5T^{2} \)
83 \( 1 - 930.T + 5.71e5T^{2} \)
89 \( 1 + 732.T + 7.04e5T^{2} \)
97 \( 1 - 1.17e3T + 9.12e5T^{2} \)
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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

−9.505782475965799818981688024496, −8.617583193755793858561833080536, −7.31502066530146943600225182901, −6.50080582693094776380446550668, −5.95465640471383046372825201340, −4.68581540727749609673383706107, −3.87565372070342882810123320050, −3.02134966522671574925917699623, −2.27134909059286964123138335012, 0, 2.27134909059286964123138335012, 3.02134966522671574925917699623, 3.87565372070342882810123320050, 4.68581540727749609673383706107, 5.95465640471383046372825201340, 6.50080582693094776380446550668, 7.31502066530146943600225182901, 8.617583193755793858561833080536, 9.505782475965799818981688024496

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