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Properties

Label 2-825-11.3-c1-0-1
Degree $2$
Conductor $825$
Sign $-0.394 + 0.918i$
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  + (−0.755 + 0.548i)2-s + (0.309 + 0.951i)3-s + (−0.348 + 1.07i)4-s + (−0.755 − 0.548i)6-s + (−1.37 + 4.22i)7-s + (−0.902 − 2.77i)8-s + (−0.809 + 0.587i)9-s + (−2.54 − 2.12i)11-s − 1.12·12-s + (−1.15 + 0.836i)13-s + (−1.28 − 3.94i)14-s + (0.379 + 0.275i)16-s + (2.09 + 1.52i)17-s + (0.288 − 0.887i)18-s + (−0.637 − 1.96i)19-s + ⋯
L(s)  = 1  + (−0.534 + 0.388i)2-s + (0.178 + 0.549i)3-s + (−0.174 + 0.536i)4-s + (−0.308 − 0.224i)6-s + (−0.519 + 1.59i)7-s + (−0.319 − 0.982i)8-s + (−0.269 + 0.195i)9-s + (−0.767 − 0.641i)11-s − 0.325·12-s + (−0.319 + 0.231i)13-s + (−0.342 − 1.05i)14-s + (0.0949 + 0.0689i)16-s + (0.508 + 0.369i)17-s + (0.0679 − 0.209i)18-s + (−0.146 − 0.449i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.394 + 0.918i$
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: \((2,\ 825,\ (\ :1/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.179331 - 0.272196i\)
\(L(\frac12)\) \(\approx\) \(0.179331 - 0.272196i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 2.12i)T \)
good2 \( 1 + (0.755 - 0.548i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (1.37 - 4.22i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.15 - 0.836i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.09 - 1.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.637 + 1.96i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 + (-1.24 + 3.84i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.87 + 2.08i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.96 - 9.13i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.531 + 1.63i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.46T + 43T^{2} \)
47 \( 1 + (-0.593 - 1.82i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.25 + 2.36i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.57 - 7.91i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.6 + 7.70i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 + (-5.61 - 4.08i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.89 + 8.89i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.87 - 2.08i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.78 + 7.10i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (4.20 - 3.05i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.49882116631269099830826487925, −9.697364644191731254645026380981, −9.041842052582238817040875280829, −8.358327932564056126868612792161, −7.74652934971190150602599543122, −6.38857654092920121341878830917, −5.71556464409172673288415389939, −4.54725117436826777903142624932, −3.28478446277993435320023606637, −2.52128448881241703046231200912, 0.18799203697493201894064083794, 1.44585989948629794925408964740, 2.74860819223019659657574890554, 4.07012117282162701106235807149, 5.19753567177490911850628518371, 6.25543386315002991728014799998, 7.33469404619179196211213499105, 7.79528013611406119090457035042, 8.930136250859372609691710323536, 9.931180002834736779718601414432

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