Properties

Label 2-4650-1.1-c1-0-76
Degree $2$
Conductor $4650$
Sign $-1$
Analytic cond. $37.1304$
Root an. cond. $6.09347$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 2.37·7-s − 8-s + 9-s + 6.37·11-s − 12-s + 2·13-s − 2.37·14-s + 16-s − 6.74·17-s − 18-s − 6.37·19-s − 2.37·21-s − 6.37·22-s + 2.37·23-s + 24-s − 2·26-s − 27-s + 2.37·28-s + 2.74·29-s − 31-s − 32-s − 6.37·33-s + 6.74·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.896·7-s − 0.353·8-s + 0.333·9-s + 1.92·11-s − 0.288·12-s + 0.554·13-s − 0.634·14-s + 0.250·16-s − 1.63·17-s − 0.235·18-s − 1.46·19-s − 0.517·21-s − 1.35·22-s + 0.494·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.448·28-s + 0.509·29-s − 0.179·31-s − 0.176·32-s − 1.10·33-s + 1.15·34-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ -1)\)

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 \( 1 \)
31 \( 1 + T \)
good7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 6.74T + 17T^{2} \)
19 \( 1 + 6.37T + 19T^{2} \)
23 \( 1 - 2.37T + 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 6.37T + 43T^{2} \)
47 \( 1 + 4.74T + 47T^{2} \)
53 \( 1 - 4.37T + 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 0.744T + 67T^{2} \)
71 \( 1 + 2.37T + 71T^{2} \)
73 \( 1 + 9.11T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.187739174110367421553311030345, −6.94043384359763290796180965102, −6.68804144562419707163809570057, −6.05975329508170626646811782590, −4.86624348356107088704035931927, −4.32630470800383424636289952301, −3.39198590475853754033954296876, −1.85955530219704585871673093711, −1.48270525370762976810609932868, 0, 1.48270525370762976810609932868, 1.85955530219704585871673093711, 3.39198590475853754033954296876, 4.32630470800383424636289952301, 4.86624348356107088704035931927, 6.05975329508170626646811782590, 6.68804144562419707163809570057, 6.94043384359763290796180965102, 8.187739174110367421553311030345

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