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Properties

Label 2-825-5.4-c3-0-1
Degree $2$
Conductor $825$
Sign $-0.894 + 0.447i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.42i·2-s − 3i·3-s − 11.5·4-s + 13.2·6-s − 31.6i·7-s − 15.8i·8-s − 9·9-s − 11·11-s + 34.7i·12-s + 5.15i·13-s + 140.·14-s − 22.6·16-s − 121. i·17-s − 39.8i·18-s − 34.8·19-s + ⋯
L(s)  = 1  + 1.56i·2-s − 0.577i·3-s − 1.44·4-s + 0.903·6-s − 1.71i·7-s − 0.699i·8-s − 0.333·9-s − 0.301·11-s + 0.835i·12-s + 0.109i·13-s + 2.67·14-s − 0.353·16-s − 1.73i·17-s − 0.521i·18-s − 0.420·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2526208087\)
\(L(\frac12)\) \(\approx\) \(0.2526208087\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 - 4.42iT - 8T^{2} \)
7 \( 1 + 31.6iT - 343T^{2} \)
13 \( 1 - 5.15iT - 2.19e3T^{2} \)
17 \( 1 + 121. iT - 4.91e3T^{2} \)
19 \( 1 + 34.8T + 6.85e3T^{2} \)
23 \( 1 - 116. iT - 1.21e4T^{2} \)
29 \( 1 - 69.4T + 2.43e4T^{2} \)
31 \( 1 - 140.T + 2.97e4T^{2} \)
37 \( 1 - 420. iT - 5.06e4T^{2} \)
41 \( 1 + 322.T + 6.89e4T^{2} \)
43 \( 1 - 321. iT - 7.95e4T^{2} \)
47 \( 1 - 231. iT - 1.03e5T^{2} \)
53 \( 1 - 4.91iT - 1.48e5T^{2} \)
59 \( 1 + 406.T + 2.05e5T^{2} \)
61 \( 1 + 556.T + 2.26e5T^{2} \)
67 \( 1 + 84.7iT - 3.00e5T^{2} \)
71 \( 1 - 49.0T + 3.57e5T^{2} \)
73 \( 1 - 785. iT - 3.89e5T^{2} \)
79 \( 1 - 383.T + 4.93e5T^{2} \)
83 \( 1 + 930. iT - 5.71e5T^{2} \)
89 \( 1 - 732.T + 7.04e5T^{2} \)
97 \( 1 - 1.17e3iT - 9.12e5T^{2} \)
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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

−10.08450344796862879208057354508, −9.270028235137935150091674837599, −8.070049042856974746795215566298, −7.66349078295702602920346070872, −6.87090082648338106374639347948, −6.40352955534580610646980517930, −5.09460488016520987726379059902, −4.46336118178523952277968438202, −3.03332406419574166703074924222, −1.15557914797694368133464218840, 0.07088090112064358365883675183, 1.85713428132662860627649155592, 2.56682912881532821197533110756, 3.56347057291526263799893827187, 4.54822864837259315271261709495, 5.55016951689399035500583943240, 6.44196288977741245650399220509, 8.284862251856012409752840337681, 8.749123771787578458202691093280, 9.513658972085820765221606546759

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