Properties

Label 2-825-5.4-c1-0-11
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  − 0.414i·2-s i·3-s + 1.82·4-s − 0.414·6-s + 4.82i·7-s − 1.58i·8-s − 9-s − 11-s − 1.82i·12-s + 5.65i·13-s + 1.99·14-s + 3·16-s + 6.82i·17-s + 0.414i·18-s + 1.17·19-s + ⋯
L(s)  = 1  − 0.292i·2-s − 0.577i·3-s + 0.914·4-s − 0.169·6-s + 1.82i·7-s − 0.560i·8-s − 0.333·9-s − 0.301·11-s − 0.527i·12-s + 1.56i·13-s + 0.534·14-s + 0.750·16-s + 1.65i·17-s + 0.0976i·18-s + 0.268·19-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

\(L(1)\) \(\approx\) \(1.75759 + 0.414910i\)
\(L(\frac12)\) \(\approx\) \(1.75759 + 0.414910i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414iT - 2T^{2} \)
7 \( 1 - 4.82iT - 7T^{2} \)
13 \( 1 - 5.65iT - 13T^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 0.343iT - 37T^{2} \)
41 \( 1 + 0.828T + 41T^{2} \)
43 \( 1 + 3.17iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 0.343T + 61T^{2} \)
67 \( 1 + 5.65iT - 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 + 0.343iT - 97T^{2} \)
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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

−10.43969702223941326611119859392, −9.339098454120590056275019911476, −8.599341278597317257799094183927, −7.79476598908460522041186935245, −6.49915623841898382295793606365, −6.25714136758179875364443566652, −5.13321722628245346220217454604, −3.58912204404934569321063156039, −2.30864727305991836298526744164, −1.84308860943031413382342096906, 0.885607331966827786438884359069, 2.80180300784318289124310698989, 3.63197644658420660360651121604, 4.90143341949170828001505059695, 5.69058803795240891829749169166, 6.95885925193177750885125328827, 7.48775495463780875543976542710, 8.151664135372379631135883524660, 9.630992226908692761741182085803, 10.22196916025729619598586871455

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