Properties

Label 2-4830-1.1-c1-0-16
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 − 1.12·13-s − 14-s − 15-s + 16-s − 1.12·17-s − 18-s − 7.12·19-s − 20-s + 21-s − 4·22-s + 23-s − 24-s + 25-s + 1.12·26-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 + 1.20·11-s + 0.288·12-s − 0.311·13-s − 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.272·17-s − 0.235·18-s − 1.63·19-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 + 0.220·26-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

\(L(1)\) \(\approx\) \(1.682945897\)
\(L(\frac12)\) \(\approx\) \(1.682945897\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.12T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 3.12T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 3.12T + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 8.24T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 2.24T + 79T^{2} \)
83 \( 1 - 9.36T + 83T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−8.420668635267934291614446340828, −7.74966451361655079873928252860, −6.79342165795778210012529170851, −6.57317391712096615609296774585, −5.35487620067887520419879189577, −4.28549585943041613546356309092, −3.83434326156505709232274467313, −2.65241311730860050464658107083, −1.89613939614338366019281941823, −0.78215512881108325652127157897, 0.78215512881108325652127157897, 1.89613939614338366019281941823, 2.65241311730860050464658107083, 3.83434326156505709232274467313, 4.28549585943041613546356309092, 5.35487620067887520419879189577, 6.57317391712096615609296774585, 6.79342165795778210012529170851, 7.74966451361655079873928252860, 8.420668635267934291614446340828

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