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Properties

Label 2-825-55.14-c1-0-13
Degree $2$
Conductor $825$
Sign $-0.992 + 0.120i$
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  + (−1.38 − 1.90i)2-s + (−0.951 + 0.309i)3-s + (−1.09 + 3.37i)4-s + (1.90 + 1.38i)6-s + (−0.184 − 0.0598i)7-s + (3.47 − 1.12i)8-s + (0.809 − 0.587i)9-s + (−1.96 + 2.67i)11-s − 3.54i·12-s + (−0.572 − 0.787i)13-s + (0.140 + 0.433i)14-s + (−1.21 − 0.880i)16-s + (1.57 − 2.16i)17-s + (−2.24 − 0.727i)18-s + (1.71 + 5.27i)19-s + ⋯
L(s)  = 1  + (−0.979 − 1.34i)2-s + (−0.549 + 0.178i)3-s + (−0.548 + 1.68i)4-s + (0.778 + 0.565i)6-s + (−0.0695 − 0.0226i)7-s + (1.22 − 0.398i)8-s + (0.269 − 0.195i)9-s + (−0.591 + 0.806i)11-s − 1.02i·12-s + (−0.158 − 0.218i)13-s + (0.0376 + 0.115i)14-s + (−0.303 − 0.220i)16-s + (0.381 − 0.525i)17-s + (−0.528 − 0.171i)18-s + (0.393 + 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.992 + 0.120i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.992 + 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0243106 - 0.401392i\)
\(L(\frac12)\) \(\approx\) \(0.0243106 - 0.401392i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.05402801853508371230181823003, −9.412907430087091223100335314203, −8.237483814788736963543200143702, −7.65596096531158636012498642292, −6.38153215701820885658458336376, −5.17935011104941028198673720139, −4.09735872959248823342704261630, −2.93977715936778550099665622619, −1.83934146523558947615495636088, −0.33762730990386126549684471823, 1.19219844100257576039905099274, 3.20759244747952877088304066987, 5.02164745102456027184899225444, 5.51366403755449445465904685825, 6.61212633925150770792665582007, 7.07459361276504606295651357446, 8.100389221013100098991605473226, 8.682709169946513131870732838429, 9.625015129544007146610232449863, 10.38364340105787459302005070331

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