Properties

Label 2-4830-1.1-c1-0-12
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 $0$

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 − 4·11-s − 12-s + 6.74·13-s + 14-s − 15-s + 16-s − 18-s + 4.74·19-s + 20-s + 21-s + 4·22-s − 23-s + 24-s + 25-s − 6.74·26-s − 27-s − 28-s + 4.74·29-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 − 1.20·11-s − 0.288·12-s + 1.87·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s + 1.08·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 0.208·23-s + 0.204·24-s + 0.200·25-s − 1.32·26-s − 0.192·27-s − 0.188·28-s + 0.881·29-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.170466717\)
\(L(\frac12)\) \(\approx\) \(1.170466717\)
\(L(\frac{3}{2})\) 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 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
29 \( 1 - 4.74T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 8.74T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 0.744T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 1.25T + 97T^{2} \)
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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

−8.362967599396316486829863472358, −7.63434123589094092818093538131, −6.74454345873079194795529448422, −6.17685834616581263875829805446, −5.55558769398112576227433633561, −4.76968200226925766138049473349, −3.54531849207073823128693323875, −2.82252417428467463945743014812, −1.63168338707558880328145729319, −0.71065407895484290259084053375, 0.71065407895484290259084053375, 1.63168338707558880328145729319, 2.82252417428467463945743014812, 3.54531849207073823128693323875, 4.76968200226925766138049473349, 5.55558769398112576227433633561, 6.17685834616581263875829805446, 6.74454345873079194795529448422, 7.63434123589094092818093538131, 8.362967599396316486829863472358

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