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Properties

Label 2-825-165.47-c1-0-53
Degree $2$
Conductor $825$
Sign $0.915 + 0.403i$
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  + (−0.0347 + 0.219i)2-s + (1.31 − 1.13i)3-s + (1.85 + 0.602i)4-s + (0.202 + 0.326i)6-s + (2.12 − 1.08i)7-s + (−0.397 + 0.780i)8-s + (0.435 − 2.96i)9-s + (−1.73 + 2.82i)11-s + (3.11 − 1.31i)12-s + (0.570 − 3.59i)13-s + (0.163 + 0.502i)14-s + (2.99 + 2.17i)16-s + (−4.30 + 0.681i)17-s + (0.635 + 0.198i)18-s + (6.07 − 1.97i)19-s + ⋯
L(s)  = 1  + (−0.0245 + 0.154i)2-s + (0.756 − 0.653i)3-s + (0.927 + 0.301i)4-s + (0.0827 + 0.133i)6-s + (0.801 − 0.408i)7-s + (−0.140 + 0.276i)8-s + (0.145 − 0.989i)9-s + (−0.522 + 0.852i)11-s + (0.898 − 0.378i)12-s + (0.158 − 0.998i)13-s + (0.0436 + 0.134i)14-s + (0.749 + 0.544i)16-s + (−1.04 + 0.165i)17-s + (0.149 + 0.0467i)18-s + (1.39 − 0.452i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.915 + 0.403i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.915 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53107 - 0.532928i\)
\(L(\frac12)\) \(\approx\) \(2.53107 - 0.532928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.31 + 1.13i)T \)
5 \( 1 \)
11 \( 1 + (1.73 - 2.82i)T \)
good2 \( 1 + (0.0347 - 0.219i)T + (-1.90 - 0.618i)T^{2} \)
7 \( 1 + (-2.12 + 1.08i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (-0.570 + 3.59i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (4.30 - 0.681i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-6.07 + 1.97i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.82 - 1.82i)T + 23iT^{2} \)
29 \( 1 + (-1.57 + 4.84i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.59 - 5.51i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.54 - 1.29i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-8.60 + 2.79i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (5.81 - 5.81i)T - 43iT^{2} \)
47 \( 1 + (-2.20 - 1.12i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (3.09 + 0.490i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (0.394 - 1.21i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.05 - 0.767i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.43 + 2.43i)T + 67iT^{2} \)
71 \( 1 + (3.81 - 5.25i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.910 - 1.78i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (2.94 + 4.05i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.41 + 15.2i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + 1.42T + 89T^{2} \)
97 \( 1 + (-0.832 - 0.131i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.23203453730695684016577297492, −9.141958160128051363834071301919, −8.189033362111093923754771133772, −7.48220554823438484066839451527, −7.15093017451760874428232465267, −5.97192694649197143564537752856, −4.80740753328135227882500433477, −3.41927298315182358844441310632, −2.48849855779990503564143310755, −1.41155653952389211361870932216, 1.70795248240055115464940384093, 2.64897701316788475105295140012, 3.69121759558970707905989461777, 4.95382588819967442460985823863, 5.73649282136040186793627599003, 6.98470192759416327375082919657, 7.79431632567976391467851629476, 8.726142658105915768093067465796, 9.364367943136221783291018548136, 10.36767970289405512758812753905

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