Properties

Label 2-825-5.4-c1-0-16
Degree $2$
Conductor $825$
Sign $0.894 - 0.447i$
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  + 2.41i·2-s i·3-s − 3.82·4-s + 2.41·6-s − 0.828i·7-s − 4.41i·8-s − 9-s − 11-s + 3.82i·12-s − 5.65i·13-s + 1.99·14-s + 2.99·16-s + 1.17i·17-s − 2.41i·18-s + 6.82·19-s + ⋯
L(s)  = 1  + 1.70i·2-s − 0.577i·3-s − 1.91·4-s + 0.985·6-s − 0.313i·7-s − 1.56i·8-s − 0.333·9-s − 0.301·11-s + 1.10i·12-s − 1.56i·13-s + 0.534·14-s + 0.749·16-s + 0.284i·17-s − 0.569i·18-s + 1.56·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17670 + 0.277781i\)
\(L(\frac12)\) \(\approx\) \(1.17670 + 0.277781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
11 \( 1 + T \)
good2 \( 1 - 2.41iT - 2T^{2} \)
7 \( 1 + 0.828iT - 7T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 - 1.17iT - 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 11.6iT - 37T^{2} \)
41 \( 1 - 4.82T + 41T^{2} \)
43 \( 1 + 8.82iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 9.31iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 5.65iT - 67T^{2} \)
71 \( 1 - 2.34T + 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + 3.65T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−10.10529241128575786041454706664, −9.022766228414750512392367202088, −8.245569449669501105313673248466, −7.51734065113180192976087766273, −7.08290136686817891922072277247, −5.82353290354193754799186635818, −5.51654273147245108412606694146, −4.28667896326212264251382151474, −2.86614929481150183736850067960, −0.67388100516465070142553089686, 1.35448624701722751053163654461, 2.62869469164556863918922539673, 3.48289237396234181886523872791, 4.51050717951824902994725217071, 5.22276486110793439709076503918, 6.61953017175350998789262794380, 7.951175351512570473982404454002, 9.072343170812401081747955661232, 9.507762931453411854010150708563, 10.16548877414478517720417166227

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