Properties

Label 2-825-1.1-c3-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.32·2-s + 3·3-s − 2.57·4-s + 6.98·6-s − 22.4·7-s − 24.6·8-s + 9·9-s + 11·11-s − 7.72·12-s + 9.86·13-s − 52.3·14-s − 36.7·16-s + 128.·17-s + 20.9·18-s + 7.04·19-s − 67.4·21-s + 25.6·22-s − 0.654·23-s − 73.8·24-s + 22.9·26-s + 27·27-s + 57.8·28-s − 229.·29-s + 155.·31-s + 111.·32-s + 33·33-s + 298.·34-s + ⋯
L(s)  = 1  + 0.823·2-s + 0.577·3-s − 0.321·4-s + 0.475·6-s − 1.21·7-s − 1.08·8-s + 0.333·9-s + 0.301·11-s − 0.185·12-s + 0.210·13-s − 0.998·14-s − 0.574·16-s + 1.82·17-s + 0.274·18-s + 0.0850·19-s − 0.700·21-s + 0.248·22-s − 0.00593·23-s − 0.628·24-s + 0.173·26-s + 0.192·27-s + 0.390·28-s − 1.46·29-s + 0.902·31-s + 0.615·32-s + 0.174·33-s + 1.50·34-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: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.840764868\)
\(L(\frac12)\) \(\approx\) \(2.840764868\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.32T + 8T^{2} \)
7 \( 1 + 22.4T + 343T^{2} \)
13 \( 1 - 9.86T + 2.19e3T^{2} \)
17 \( 1 - 128.T + 4.91e3T^{2} \)
19 \( 1 - 7.04T + 6.85e3T^{2} \)
23 \( 1 + 0.654T + 1.21e4T^{2} \)
29 \( 1 + 229.T + 2.43e4T^{2} \)
31 \( 1 - 155.T + 2.97e4T^{2} \)
37 \( 1 - 110.T + 5.06e4T^{2} \)
41 \( 1 - 154.T + 6.89e4T^{2} \)
43 \( 1 - 401.T + 7.95e4T^{2} \)
47 \( 1 - 277.T + 1.03e5T^{2} \)
53 \( 1 - 651.T + 1.48e5T^{2} \)
59 \( 1 + 423.T + 2.05e5T^{2} \)
61 \( 1 - 681.T + 2.26e5T^{2} \)
67 \( 1 + 374.T + 3.00e5T^{2} \)
71 \( 1 - 96.6T + 3.57e5T^{2} \)
73 \( 1 - 19.9T + 3.89e5T^{2} \)
79 \( 1 - 24.4T + 4.93e5T^{2} \)
83 \( 1 - 1.12e3T + 5.71e5T^{2} \)
89 \( 1 + 639.T + 7.04e5T^{2} \)
97 \( 1 - 730.T + 9.12e5T^{2} \)
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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

−9.566689025189692554239618416207, −9.260930493310804038584174733051, −8.141253089791682904017279604648, −7.19538017647700608216589101276, −6.08708190831412417499210704726, −5.48565476795241888890576898314, −4.12390094827828533918595865420, −3.51096882456687860401636107810, −2.66408071374131149618080691913, −0.812636248968871468148556731499, 0.812636248968871468148556731499, 2.66408071374131149618080691913, 3.51096882456687860401636107810, 4.12390094827828533918595865420, 5.48565476795241888890576898314, 6.08708190831412417499210704726, 7.19538017647700608216589101276, 8.141253089791682904017279604648, 9.260930493310804038584174733051, 9.566689025189692554239618416207

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