Properties

Label 2-4650-1.1-c1-0-62
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 + 3·7-s + 8-s + 9-s + 3·11-s + 12-s + 2·13-s + 3·14-s + 16-s + 4·17-s + 18-s − 3·19-s + 3·21-s + 3·22-s − 5·23-s + 24-s + 2·26-s + 27-s + 3·28-s + 4·29-s + 31-s + 32-s + 3·33-s + 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.801·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.688·19-s + 0.654·21-s + 0.639·22-s − 1.04·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.566·28-s + 0.742·29-s + 0.179·31-s + 0.176·32-s + 0.522·33-s + 0.685·34-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: $\chi_{4650} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.043265576\)
\(L(\frac12)\) \(\approx\) \(5.043265576\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.179465993255970405884074967119, −7.74153241357675715017471088373, −6.74551289508181603008919709039, −6.13324876690007371037739885009, −5.26698158762698358805063469928, −4.42776683955685438460685738324, −3.89612651996431308605905760703, −3.01714077393479385545797310097, −1.94548506923537729035949511135, −1.24928453466983135859884605139, 1.24928453466983135859884605139, 1.94548506923537729035949511135, 3.01714077393479385545797310097, 3.89612651996431308605905760703, 4.42776683955685438460685738324, 5.26698158762698358805063469928, 6.13324876690007371037739885009, 6.74551289508181603008919709039, 7.74153241357675715017471088373, 8.179465993255970405884074967119

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