Properties

Label 2-4030-1.1-c1-0-13
Degree $2$
Conductor $4030$
Sign $1$
Analytic cond. $32.1797$
Root an. cond. $5.67271$
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 + 1.21·3-s + 4-s − 5-s − 1.21·6-s − 1.25·7-s − 8-s − 1.51·9-s + 10-s − 6.19·11-s + 1.21·12-s + 13-s + 1.25·14-s − 1.21·15-s + 16-s + 3.94·17-s + 1.51·18-s − 4.85·19-s − 20-s − 1.53·21-s + 6.19·22-s + 8.13·23-s − 1.21·24-s + 25-s − 26-s − 5.50·27-s − 1.25·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.704·3-s + 0.5·4-s − 0.447·5-s − 0.498·6-s − 0.474·7-s − 0.353·8-s − 0.503·9-s + 0.316·10-s − 1.86·11-s + 0.352·12-s + 0.277·13-s + 0.335·14-s − 0.314·15-s + 0.250·16-s + 0.956·17-s + 0.356·18-s − 1.11·19-s − 0.223·20-s − 0.334·21-s + 1.32·22-s + 1.69·23-s − 0.249·24-s + 0.200·25-s − 0.196·26-s − 1.05·27-s − 0.237·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(32.1797\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4030,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9492809077\)
\(L(\frac12)\) \(\approx\) \(0.9492809077\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 - T \)
31 \( 1 + T \)
good3 \( 1 - 1.21T + 3T^{2} \)
7 \( 1 + 1.25T + 7T^{2} \)
11 \( 1 + 6.19T + 11T^{2} \)
17 \( 1 - 3.94T + 17T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
23 \( 1 - 8.13T + 23T^{2} \)
29 \( 1 - 0.0979T + 29T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 + 3.70T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 4.79T + 47T^{2} \)
53 \( 1 - 3.74T + 53T^{2} \)
59 \( 1 + 5.37T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 7.70T + 67T^{2} \)
71 \( 1 + 2.68T + 71T^{2} \)
73 \( 1 + 0.389T + 73T^{2} \)
79 \( 1 - 1.52T + 79T^{2} \)
83 \( 1 - 6.14T + 83T^{2} \)
89 \( 1 + 1.25T + 89T^{2} \)
97 \( 1 - 7.44T + 97T^{2} \)
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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

−8.480737951906342058215653622317, −7.82863494154161212809758345710, −7.33934862680736209964182165871, −6.35840706220335313281817201904, −5.54119263750002512309547247311, −4.71928200653563199564731549349, −3.37415458604777810592516859172, −2.97176165961672960544206391017, −2.11093688118345606583694039324, −0.56605881424058590629518601348, 0.56605881424058590629518601348, 2.11093688118345606583694039324, 2.97176165961672960544206391017, 3.37415458604777810592516859172, 4.71928200653563199564731549349, 5.54119263750002512309547247311, 6.35840706220335313281817201904, 7.33934862680736209964182165871, 7.82863494154161212809758345710, 8.480737951906342058215653622317

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