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Properties

Label 2-825-5.4-c3-0-71
Degree $2$
Conductor $825$
Sign $0.447 + 0.894i$
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  + 3.76i·2-s − 3i·3-s − 6.16·4-s + 11.2·6-s − 23.6i·7-s + 6.92i·8-s − 9·9-s + 11·11-s + 18.4i·12-s − 7.39i·13-s + 88.9·14-s − 75.3·16-s + 8.68i·17-s − 33.8i·18-s + 69.7·19-s + ⋯
L(s)  = 1  + 1.33i·2-s − 0.577i·3-s − 0.770·4-s + 0.768·6-s − 1.27i·7-s + 0.305i·8-s − 0.333·9-s + 0.301·11-s + 0.444i·12-s − 0.157i·13-s + 1.69·14-s − 1.17·16-s + 0.123i·17-s − 0.443i·18-s + 0.841·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.447 + 0.894i$
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: \((2,\ 825,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.127066155\)
\(L(\frac12)\) \(\approx\) \(1.127066155\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.76iT - 8T^{2} \)
7 \( 1 + 23.6iT - 343T^{2} \)
13 \( 1 + 7.39iT - 2.19e3T^{2} \)
17 \( 1 - 8.68iT - 4.91e3T^{2} \)
19 \( 1 - 69.7T + 6.85e3T^{2} \)
23 \( 1 - 10.1iT - 1.21e4T^{2} \)
29 \( 1 - 73.2T + 2.43e4T^{2} \)
31 \( 1 + 290.T + 2.97e4T^{2} \)
37 \( 1 + 105. iT - 5.06e4T^{2} \)
41 \( 1 - 40.5T + 6.89e4T^{2} \)
43 \( 1 - 77.3iT - 7.95e4T^{2} \)
47 \( 1 + 472. iT - 1.03e5T^{2} \)
53 \( 1 + 205. iT - 1.48e5T^{2} \)
59 \( 1 + 330.T + 2.05e5T^{2} \)
61 \( 1 + 931.T + 2.26e5T^{2} \)
67 \( 1 + 418. iT - 3.00e5T^{2} \)
71 \( 1 + 506.T + 3.57e5T^{2} \)
73 \( 1 + 612. iT - 3.89e5T^{2} \)
79 \( 1 + 54.6T + 4.93e5T^{2} \)
83 \( 1 + 538. iT - 5.71e5T^{2} \)
89 \( 1 + 781.T + 7.04e5T^{2} \)
97 \( 1 + 531. iT - 9.12e5T^{2} \)
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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

−9.408623200048095539240105411495, −8.532465060340582297005021951086, −7.48940355416636595629533043321, −7.32713528785550629419616750014, −6.38595016602549267152879380498, −5.54819137970966928357043244147, −4.52412923664882846357759893057, −3.34303868774747218845094408911, −1.69050342382578813827057728800, −0.29566007449265509861407278872, 1.36581903008308745883066158917, 2.52093769019070613527859655370, 3.27066285531628959055683253862, 4.33240884004372481695020884301, 5.34653833215934008842004664501, 6.30018635421760290657145405397, 7.55828581792002651017029381353, 8.912696850148006632482775140623, 9.222550893188483471796094144801, 10.05323418496348587551691062002

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