Properties

Label 2-8030-1.1-c1-0-141
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 + 0.335·3-s + 4-s + 5-s − 0.335·6-s − 3.89·7-s − 8-s − 2.88·9-s − 10-s − 11-s + 0.335·12-s + 1.74·13-s + 3.89·14-s + 0.335·15-s + 16-s + 0.959·17-s + 2.88·18-s − 0.980·19-s + 20-s − 1.30·21-s + 22-s + 6.94·23-s − 0.335·24-s + 25-s − 1.74·26-s − 1.97·27-s − 3.89·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.193·3-s + 0.5·4-s + 0.447·5-s − 0.137·6-s − 1.47·7-s − 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.301·11-s + 0.0969·12-s + 0.484·13-s + 1.04·14-s + 0.0867·15-s + 0.250·16-s + 0.232·17-s + 0.680·18-s − 0.224·19-s + 0.223·20-s − 0.285·21-s + 0.213·22-s + 1.44·23-s − 0.0685·24-s + 0.200·25-s − 0.342·26-s − 0.380·27-s − 0.736·28-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 \( 1 - 0.335T + 3T^{2} \)
7 \( 1 + 3.89T + 7T^{2} \)
13 \( 1 - 1.74T + 13T^{2} \)
17 \( 1 - 0.959T + 17T^{2} \)
19 \( 1 + 0.980T + 19T^{2} \)
23 \( 1 - 6.94T + 23T^{2} \)
29 \( 1 + 4.84T + 29T^{2} \)
31 \( 1 - 5.52T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 - 3.60T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 0.509T + 47T^{2} \)
53 \( 1 + 13.9T + 53T^{2} \)
59 \( 1 - 8.18T + 59T^{2} \)
61 \( 1 + 1.07T + 61T^{2} \)
67 \( 1 + 5.45T + 67T^{2} \)
71 \( 1 + 1.51T + 71T^{2} \)
79 \( 1 - 1.12T + 79T^{2} \)
83 \( 1 - 6.23T + 83T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−7.57063631790780722631021596239, −6.71881776495209868961910598642, −6.21716047811459154189449847602, −5.70395086005260243458417585631, −4.75275689041625063775422562991, −3.50228971476183962734292657726, −3.00719404703916640093969032845, −2.35316270191351374435935614608, −1.07706981266973021538952561315, 0, 1.07706981266973021538952561315, 2.35316270191351374435935614608, 3.00719404703916640093969032845, 3.50228971476183962734292657726, 4.75275689041625063775422562991, 5.70395086005260243458417585631, 6.21716047811459154189449847602, 6.71881776495209868961910598642, 7.57063631790780722631021596239

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