Properties

Label 2-6600-5.4-c1-0-23
Degree $2$
Conductor $6600$
Sign $0.447 - 0.894i$
Analytic cond. $52.7012$
Root an. cond. $7.25956$
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  i·3-s + 4i·7-s − 9-s − 11-s − 6i·13-s + 6i·17-s + 8·19-s + 4·21-s + i·27-s + 6·29-s + i·33-s + 6i·37-s − 6·39-s − 10·41-s + 8i·43-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.51i·7-s − 0.333·9-s − 0.301·11-s − 1.66i·13-s + 1.45i·17-s + 1.83·19-s + 0.872·21-s + 0.192i·27-s + 1.11·29-s + 0.174i·33-s + 0.986i·37-s − 0.960·39-s − 1.56·41-s + 1.21i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(52.7012\)
Root analytic conductor: \(7.25956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6600} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6600,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.657791872\)
\(L(\frac12)\) \(\approx\) \(1.657791872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 - 4iT - 7T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.189895042921327216937712360855, −7.63346943361923356497698039418, −6.55932841480255888887304010300, −5.98904602418638324469940649485, −5.37559866717523881095622124380, −4.85794366061380761291505487445, −3.25660479273900506999309294066, −3.05210487621140954362395056101, −1.99033277102822886095514260343, −1.02953379901027700480141895230, 0.46968592589364323727335755822, 1.51709663962674736945583071088, 2.78840941093918128962596917861, 3.54733043450864466714919115250, 4.32751423961662575011243128347, 4.83360876791575617152765891508, 5.59417785446494961163652515086, 6.73579524257721472130114487868, 7.16790271110017553532003256129, 7.65013478411981669343286991700

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