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Properties

Label 2-825-11.3-c1-0-3
Degree $2$
Conductor $825$
Sign $-0.941 - 0.335i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.90 + 1.38i)2-s + (−0.309 − 0.951i)3-s + (1.09 − 3.37i)4-s + (1.90 + 1.38i)6-s + (−0.0598 + 0.184i)7-s + (1.12 + 3.47i)8-s + (−0.809 + 0.587i)9-s + (−1.96 + 2.67i)11-s − 3.54·12-s + (0.787 − 0.572i)13-s + (−0.140 − 0.433i)14-s + (−1.21 − 0.880i)16-s + (−2.16 − 1.57i)17-s + (0.727 − 2.24i)18-s + (−1.71 − 5.27i)19-s + ⋯
L(s)  = 1  + (−1.34 + 0.979i)2-s + (−0.178 − 0.549i)3-s + (0.548 − 1.68i)4-s + (0.778 + 0.565i)6-s + (−0.0226 + 0.0695i)7-s + (0.398 + 1.22i)8-s + (−0.269 + 0.195i)9-s + (−0.591 + 0.806i)11-s − 1.02·12-s + (0.218 − 0.158i)13-s + (−0.0376 − 0.115i)14-s + (−0.303 − 0.220i)16-s + (−0.525 − 0.381i)17-s + (0.171 − 0.528i)18-s + (−0.393 − 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.941 - 0.335i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.941 - 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0473764 + 0.273807i\)
\(L(\frac12)\) \(\approx\) \(0.0473764 + 0.273807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
11 \( 1 + (1.96 - 2.67i)T \)
good2 \( 1 + (1.90 - 1.38i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.0598 - 0.184i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.787 + 0.572i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.16 + 1.57i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.71 + 5.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.80T + 23T^{2} \)
29 \( 1 + (3.12 - 9.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.02 + 1.47i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.76 - 5.43i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.55 + 7.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.11T + 43T^{2} \)
47 \( 1 + (-3.35 - 10.3i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.51 - 5.45i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.46 - 10.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.975 + 0.708i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.25T + 67T^{2} \)
71 \( 1 + (4.84 + 3.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.02 - 3.14i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.21 - 5.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.72 - 4.88i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.34T + 89T^{2} \)
97 \( 1 + (12.8 - 9.31i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.59381658095182490006334989200, −9.342216835822597277296742317800, −8.921106422033149051170588464266, −7.982115138846019221313784756610, −7.10652607783213400046811371488, −6.80462397004922640419353192803, −5.64597206641818852264831876897, −4.73361111262878793668951464737, −2.75357707938708358230377679900, −1.31947293072393905321363599341, 0.23109309044442419498346713494, 1.83961600834612299699993053991, 3.06703253614048790386876383523, 3.99799153778790712354328933976, 5.40155163616677505317295683993, 6.46832889205088661291143283134, 7.79202038170659193927868146267, 8.418886280652792730606508708061, 9.117831169477677455098861026415, 10.03559324947811365943059954819

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