Properties

Label 2-4650-1.1-c1-0-22
Degree $2$
Conductor $4650$
Sign $1$
Analytic cond. $37.1304$
Root an. cond. $6.09347$
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 − 11-s + 12-s − 13-s − 14-s + 16-s + 6·17-s − 18-s + 21-s + 22-s − 24-s + 26-s + 27-s + 28-s + 6·29-s + 31-s − 32-s − 33-s − 6·34-s + 36-s − 5·37-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 − 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.218·21-s + 0.213·22-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.179·31-s − 0.176·32-s − 0.174·33-s − 1.02·34-s + 1/6·36-s − 0.821·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 31\)
Sign: $1$
Analytic conductor: \(37.1304\)
Root analytic conductor: \(6.09347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.879006358\)
\(L(\frac12)\) \(\approx\) \(1.879006358\)
\(L(\frac{3}{2})\) 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 \( 1 \)
31 \( 1 - T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
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

−8.171212384042543086064243682088, −7.83184202406839612692424327398, −7.12294714368617492224790851622, −6.27793805699053861891692550187, −5.39518024413601363623191450949, −4.61249338545131996728685597096, −3.51740322183097518164299590574, −2.80318452248472544711549951849, −1.85305167374744620205957579310, −0.846558027478012916629995258008, 0.846558027478012916629995258008, 1.85305167374744620205957579310, 2.80318452248472544711549951849, 3.51740322183097518164299590574, 4.61249338545131996728685597096, 5.39518024413601363623191450949, 6.27793805699053861891692550187, 7.12294714368617492224790851622, 7.83184202406839612692424327398, 8.171212384042543086064243682088

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