Properties

Label 2-4830-1.1-c1-0-34
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 − 2.94·11-s − 12-s + 1.28·13-s + 14-s − 15-s + 16-s + 4.22·17-s + 18-s + 6.22·19-s + 20-s − 21-s − 2.94·22-s − 23-s − 24-s + 25-s + 1.28·26-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 − 0.888·11-s − 0.288·12-s + 0.356·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 1.02·17-s + 0.235·18-s + 1.42·19-s + 0.223·20-s − 0.218·21-s − 0.628·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.251·26-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.094805627\)
\(L(\frac12)\) \(\approx\) \(3.094805627\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.94T + 11T^{2} \)
13 \( 1 - 1.28T + 13T^{2} \)
17 \( 1 - 4.22T + 17T^{2} \)
19 \( 1 - 6.22T + 19T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 + 2.22T + 31T^{2} \)
37 \( 1 + 6.45T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 2.22T + 47T^{2} \)
53 \( 1 + 6.45T + 53T^{2} \)
59 \( 1 - 8.45T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2.94T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 3.43T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.79T + 83T^{2} \)
89 \( 1 - 3.17T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−8.031633336969792827908338024354, −7.48480309462912316078584363755, −6.68912461121706908581559208455, −5.89016291864801246473997044149, −5.23573169650254509806830616711, −4.93916748688378984757777106575, −3.73178600669656581546052768675, −3.01948285864853911944164422164, −1.94348553472040814541728902041, −0.935834192190042340345114247002, 0.935834192190042340345114247002, 1.94348553472040814541728902041, 3.01948285864853911944164422164, 3.73178600669656581546052768675, 4.93916748688378984757777106575, 5.23573169650254509806830616711, 5.89016291864801246473997044149, 6.68912461121706908581559208455, 7.48480309462912316078584363755, 8.031633336969792827908338024354

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