Properties

Label 2-5610-1.1-c1-0-44
Degree $2$
Conductor $5610$
Sign $1$
Analytic cond. $44.7960$
Root an. cond. $6.69298$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s + 6·13-s − 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s − 22-s − 24-s + 25-s + 6·26-s − 27-s + 6·29-s − 30-s + 32-s + 33-s + 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5610\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(44.7960\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.285680748\)
\(L(\frac12)\) \(\approx\) \(3.285680748\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.188417740517970992720829984665, −7.10921202168148421072892210789, −6.58936583006946403054117622557, −5.85845178389626729703825109153, −5.35293002200536442688490589365, −4.62145531695555075992041476111, −3.66208844590091980294358354394, −3.03745580598348365105819231017, −1.82147199045142198102286439757, −0.953539657196692295288160220238, 0.953539657196692295288160220238, 1.82147199045142198102286439757, 3.03745580598348365105819231017, 3.66208844590091980294358354394, 4.62145531695555075992041476111, 5.35293002200536442688490589365, 5.85845178389626729703825109153, 6.58936583006946403054117622557, 7.10921202168148421072892210789, 8.188417740517970992720829984665

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