Properties

Label 2-4830-1.1-c1-0-56
Degree $2$
Conductor $4830$
Sign $-1$
Analytic cond. $38.5677$
Root an. cond. $6.21029$
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 − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 2·11-s + 12-s − 2.60·13-s + 14-s − 15-s + 16-s + 4.60·17-s − 18-s − 2.60·19-s − 20-s − 21-s + 2·22-s + 23-s − 24-s + 25-s + 2.60·26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.722·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 1.11·17-s − 0.235·18-s − 0.597·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.510·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4830\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(38.5677\)
Root analytic conductor: \(6.21029\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4830} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4830,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2.60T + 13T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 + 2.60T + 19T^{2} \)
29 \( 1 - 5.21T + 29T^{2} \)
31 \( 1 - 2.60T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 1.39T + 47T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 + 5.21T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 3.21T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−8.039919680204466614987862539193, −7.32351862606737712027502723467, −6.80078597075796584936728177354, −5.81654082660843496525887630050, −4.95296779091825928960110423220, −4.01770057736007391864931177361, −3.05916569537052200155536694523, −2.51678768453414848261967850860, −1.28565551540535630306896476209, 0, 1.28565551540535630306896476209, 2.51678768453414848261967850860, 3.05916569537052200155536694523, 4.01770057736007391864931177361, 4.95296779091825928960110423220, 5.81654082660843496525887630050, 6.80078597075796584936728177354, 7.32351862606737712027502723467, 8.039919680204466614987862539193

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