Properties

Label 2-5550-1.1-c1-0-65
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 $0$

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 + 5·11-s + 12-s + 14-s + 16-s + 17-s + 18-s + 21-s + 5·22-s + 4·23-s + 24-s + 27-s + 28-s − 3·29-s + 31-s + 32-s + 5·33-s + 34-s + 36-s − 37-s − 41-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 + 1.50·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.218·21-s + 1.06·22-s + 0.834·23-s + 0.204·24-s + 0.192·27-s + 0.188·28-s − 0.557·29-s + 0.179·31-s + 0.176·32-s + 0.870·33-s + 0.171·34-s + 1/6·36-s − 0.164·37-s − 0.156·41-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: $\chi_{5550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.886503456\)
\(L(\frac12)\) \(\approx\) \(4.886503456\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.063592523843463910411652305095, −7.35898301680426009716170461676, −6.69909421143915294828864378679, −6.03902997875320319240871566207, −5.12260945243794199972900778980, −4.38551364064696574782080249507, −3.70981207992547093330763589254, −3.00480813102632151094402578555, −1.94987107227209405160567572860, −1.14030625268131971647013693232, 1.14030625268131971647013693232, 1.94987107227209405160567572860, 3.00480813102632151094402578555, 3.70981207992547093330763589254, 4.38551364064696574782080249507, 5.12260945243794199972900778980, 6.03902997875320319240871566207, 6.69909421143915294828864378679, 7.35898301680426009716170461676, 8.063592523843463910411652305095

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