Properties

Label 2-825-5.4-c1-0-27
Degree $2$
Conductor $825$
Sign $-0.447 - 0.894i$
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  − 2.12i·2-s i·3-s − 2.51·4-s − 2.12·6-s − 3.64i·7-s + 1.09i·8-s − 9-s + 11-s + 2.51i·12-s + 1.51i·13-s − 7.73·14-s − 2.70·16-s − 1.15i·17-s + 2.12i·18-s − 2.60·19-s + ⋯
L(s)  = 1  − 1.50i·2-s − 0.577i·3-s − 1.25·4-s − 0.867·6-s − 1.37i·7-s + 0.387i·8-s − 0.333·9-s + 0.301·11-s + 0.726i·12-s + 0.420i·13-s − 2.06·14-s − 0.676·16-s − 0.280i·17-s + 0.500i·18-s − 0.598·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(6.58765\)
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: \((2,\ 825,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.604938 + 0.978810i\)
\(L(\frac12)\) \(\approx\) \(0.604938 + 0.978810i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.12iT - 2T^{2} \)
7 \( 1 + 3.64iT - 7T^{2} \)
13 \( 1 - 1.51iT - 13T^{2} \)
17 \( 1 + 1.15iT - 17T^{2} \)
19 \( 1 + 2.60T + 19T^{2} \)
23 \( 1 + 5.73iT - 23T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 5.51T + 31T^{2} \)
37 \( 1 + 0.454iT - 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 - 3.48iT - 47T^{2} \)
53 \( 1 + 12.5iT - 53T^{2} \)
59 \( 1 - 7.73T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 14.2iT - 67T^{2} \)
71 \( 1 - 8.51T + 71T^{2} \)
73 \( 1 + 9.21iT - 73T^{2} \)
79 \( 1 + 5.09T + 79T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 6.77iT - 97T^{2} \)
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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

−9.926791958087958072114896537981, −9.095859944584147206381944421647, −8.050538275006145381608630409565, −7.05315812352007219562708876129, −6.32779562786236021567732250748, −4.63145756213600638102151179305, −3.97935231045098896335378031425, −2.85306763793579772075483243932, −1.68729420040418093409655607667, −0.56744977828891505152320765071, 2.32285850768391606188067586596, 3.77309870844691475126191373343, 4.98381156796781296434293067236, 5.67981715122935171903681675289, 6.25420744733698131452314451541, 7.38926996729282910680261481348, 8.243922113188058498243258141922, 8.957452143185897948066623279841, 9.491270156066729688650409678289, 10.68654892140462194465546155143

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