Properties

Label 2-825-1.1-c3-0-14
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  + 2.56·2-s − 3·3-s − 1.43·4-s − 7.68·6-s − 6.24·7-s − 24.1·8-s + 9·9-s − 11·11-s + 4.31·12-s + 49.1·13-s − 16·14-s − 50.4·16-s − 82.7·17-s + 23.0·18-s − 130.·19-s + 18.7·21-s − 28.1·22-s + 185.·23-s + 72.5·24-s + 125.·26-s − 27·27-s + 8.98·28-s − 8.90·29-s + 5.26·31-s + 64.2·32-s + 33·33-s − 211.·34-s + ⋯
L(s)  = 1  + 0.905·2-s − 0.577·3-s − 0.179·4-s − 0.522·6-s − 0.337·7-s − 1.06·8-s + 0.333·9-s − 0.301·11-s + 0.103·12-s + 1.04·13-s − 0.305·14-s − 0.787·16-s − 1.17·17-s + 0.301·18-s − 1.57·19-s + 0.194·21-s − 0.273·22-s + 1.68·23-s + 0.616·24-s + 0.949·26-s − 0.192·27-s + 0.0606·28-s − 0.0570·29-s + 0.0304·31-s + 0.354·32-s + 0.174·33-s − 1.06·34-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: $\chi_{825} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.697234242\)
\(L(\frac12)\) \(\approx\) \(1.697234242\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.56T + 8T^{2} \)
7 \( 1 + 6.24T + 343T^{2} \)
13 \( 1 - 49.1T + 2.19e3T^{2} \)
17 \( 1 + 82.7T + 4.91e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 - 185.T + 1.21e4T^{2} \)
29 \( 1 + 8.90T + 2.43e4T^{2} \)
31 \( 1 - 5.26T + 2.97e4T^{2} \)
37 \( 1 - 416.T + 5.06e4T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 - 513.T + 7.95e4T^{2} \)
47 \( 1 + 557.T + 1.03e5T^{2} \)
53 \( 1 - 168.T + 1.48e5T^{2} \)
59 \( 1 - 618.T + 2.05e5T^{2} \)
61 \( 1 - 786.T + 2.26e5T^{2} \)
67 \( 1 - 339.T + 3.00e5T^{2} \)
71 \( 1 - 1.12e3T + 3.57e5T^{2} \)
73 \( 1 - 123.T + 3.89e5T^{2} \)
79 \( 1 + 309.T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3T + 5.71e5T^{2} \)
89 \( 1 + 141.T + 7.04e5T^{2} \)
97 \( 1 + 798.T + 9.12e5T^{2} \)
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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

−9.876528510536384947953267090958, −8.954080081240248141494269535549, −8.277786429051531621205711879598, −6.71788831850485457084068465706, −6.32318897000562416886075608313, −5.28967833031309285134040649406, −4.47846261270584869041862970750, −3.66192660395499297026734709011, −2.42307071631622425828895461978, −0.63797008900176234647170914173, 0.63797008900176234647170914173, 2.42307071631622425828895461978, 3.66192660395499297026734709011, 4.47846261270584869041862970750, 5.28967833031309285134040649406, 6.32318897000562416886075608313, 6.71788831850485457084068465706, 8.277786429051531621205711879598, 8.954080081240248141494269535549, 9.876528510536384947953267090958

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