Properties

Label 2-4830-1.1-c1-0-32
Degree $2$
Conductor $4830$
Sign $1$
Analytic cond. $38.5677$
Root an. cond. $6.21029$
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 − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s + 6·13-s + 14-s + 15-s + 16-s + 4·17-s + 18-s + 4·19-s − 20-s − 21-s + 23-s − 24-s + 25-s + 6·26-s − 27-s + 28-s − 8·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s − 1.48·29-s + 0.182·30-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: yes
Arithmetic: yes
Character: $\chi_{4830} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.857761409\)
\(L(\frac12)\) \(\approx\) \(2.857761409\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.973557986881412818836476970791, −7.56599952206944281368783277013, −6.70527290395613656515884751813, −5.87004328384711505612906765128, −5.46088457442634571949672175545, −4.60078110471312864644789352836, −3.70380113236197646640472170556, −3.27900431891209090186420252012, −1.81500177403898795118038506805, −0.920161601076446811024041427299, 0.920161601076446811024041427299, 1.81500177403898795118038506805, 3.27900431891209090186420252012, 3.70380113236197646640472170556, 4.60078110471312864644789352836, 5.46088457442634571949672175545, 5.87004328384711505612906765128, 6.70527290395613656515884751813, 7.56599952206944281368783277013, 7.973557986881412818836476970791

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