Properties

Label 2-5550-1.1-c1-0-80
Degree $2$
Conductor $5550$
Sign $-1$
Analytic cond. $44.3169$
Root an. cond. $6.65709$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 3·11-s − 12-s + 13-s − 14-s + 16-s + 3·17-s − 18-s − 7·19-s − 21-s − 3·22-s − 3·23-s + 24-s − 26-s − 27-s + 28-s + 2·31-s − 32-s − 3·33-s − 3·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.60·19-s − 0.218·21-s − 0.639·22-s − 0.625·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.359·31-s − 0.176·32-s − 0.522·33-s − 0.514·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(44.3169\)
Root analytic conductor: \(6.65709\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5550,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−8.001513662168442863978309623759, −7.01437951887825475719301902451, −6.39769483319444291966101608301, −5.90767299024270226404425939091, −4.88110072193851369524093531967, −4.14333102039663126277605679814, −3.24406126368525216574706922173, −1.98471408431458957288316935137, −1.28716270479178369718416467812, 0, 1.28716270479178369718416467812, 1.98471408431458957288316935137, 3.24406126368525216574706922173, 4.14333102039663126277605679814, 4.88110072193851369524093531967, 5.90767299024270226404425939091, 6.39769483319444291966101608301, 7.01437951887825475719301902451, 8.001513662168442863978309623759

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