Properties

Label 2-8030-1.1-c1-0-189
Degree $2$
Conductor $8030$
Sign $-1$
Analytic cond. $64.1198$
Root an. cond. $8.00748$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.882·3-s + 4-s + 5-s − 0.882·6-s + 0.390·7-s − 8-s − 2.22·9-s − 10-s − 11-s + 0.882·12-s − 0.473·13-s − 0.390·14-s + 0.882·15-s + 16-s + 3.02·17-s + 2.22·18-s − 2.37·19-s + 20-s + 0.344·21-s + 22-s + 2.87·23-s − 0.882·24-s + 25-s + 0.473·26-s − 4.60·27-s + 0.390·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.509·3-s + 0.5·4-s + 0.447·5-s − 0.360·6-s + 0.147·7-s − 0.353·8-s − 0.740·9-s − 0.316·10-s − 0.301·11-s + 0.254·12-s − 0.131·13-s − 0.104·14-s + 0.227·15-s + 0.250·16-s + 0.734·17-s + 0.523·18-s − 0.544·19-s + 0.223·20-s + 0.0751·21-s + 0.213·22-s + 0.598·23-s − 0.180·24-s + 0.200·25-s + 0.0928·26-s − 0.886·27-s + 0.0737·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8030\)    =    \(2 \cdot 5 \cdot 11 \cdot 73\)
Sign: $-1$
Analytic conductor: \(64.1198\)
Root analytic conductor: \(8.00748\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8030,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
73 \( 1 - T \)
good3 \( 1 - 0.882T + 3T^{2} \)
7 \( 1 - 0.390T + 7T^{2} \)
13 \( 1 + 0.473T + 13T^{2} \)
17 \( 1 - 3.02T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
23 \( 1 - 2.87T + 23T^{2} \)
29 \( 1 - 0.244T + 29T^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 + 7.22T + 37T^{2} \)
41 \( 1 + 0.0515T + 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + 3.31T + 47T^{2} \)
53 \( 1 + 3.47T + 53T^{2} \)
59 \( 1 + 2.93T + 59T^{2} \)
61 \( 1 - 4.36T + 61T^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 - 1.51T + 71T^{2} \)
79 \( 1 + 8.54T + 79T^{2} \)
83 \( 1 - 2.24T + 83T^{2} \)
89 \( 1 + 9.59T + 89T^{2} \)
97 \( 1 - 19.6T + 97T^{2} \)
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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

−7.69703567850880575129206139390, −6.91414017326819815042730435322, −6.18568007628712502131550214277, −5.48920891265796047799784698351, −4.80112435571109445752878432742, −3.61263675165942453348652111842, −2.92627728661700255386501682717, −2.21251161401874244472290456146, −1.31100055865281995257273272078, 0, 1.31100055865281995257273272078, 2.21251161401874244472290456146, 2.92627728661700255386501682717, 3.61263675165942453348652111842, 4.80112435571109445752878432742, 5.48920891265796047799784698351, 6.18568007628712502131550214277, 6.91414017326819815042730435322, 7.69703567850880575129206139390

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