Properties

Label 2-8030-1.1-c1-0-160
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 − 1.56·3-s + 4-s − 5-s − 1.56·6-s − 0.438·7-s + 8-s − 0.561·9-s − 10-s + 11-s − 1.56·12-s − 2·13-s − 0.438·14-s + 1.56·15-s + 16-s − 4.12·17-s − 0.561·18-s + 19-s − 20-s + 0.684·21-s + 22-s + 5.68·23-s − 1.56·24-s + 25-s − 2·26-s + 5.56·27-s − 0.438·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.901·3-s + 0.5·4-s − 0.447·5-s − 0.637·6-s − 0.165·7-s + 0.353·8-s − 0.187·9-s − 0.316·10-s + 0.301·11-s − 0.450·12-s − 0.554·13-s − 0.117·14-s + 0.403·15-s + 0.250·16-s − 0.999·17-s − 0.132·18-s + 0.229·19-s − 0.223·20-s + 0.149·21-s + 0.213·22-s + 1.18·23-s − 0.318·24-s + 0.200·25-s − 0.392·26-s + 1.07·27-s − 0.0828·28-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 \( 1 + 1.56T + 3T^{2} \)
7 \( 1 + 0.438T + 7T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 4.12T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 5.68T + 23T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 - 6.12T + 31T^{2} \)
37 \( 1 - 4.12T + 37T^{2} \)
41 \( 1 - 7.68T + 41T^{2} \)
43 \( 1 - 4.56T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 + 9.12T + 59T^{2} \)
61 \( 1 + 9.56T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + 4.87T + 97T^{2} \)
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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

−7.40688436583773574748703499008, −6.48609200975402783303901709164, −6.18576355541795905515003569702, −5.31674636481810275121662786152, −4.69979994823695534585258083865, −4.14960916083854517767813607157, −3.11607214735780044486224253512, −2.47154551228657390587241844980, −1.15285141148719897501403957951, 0, 1.15285141148719897501403957951, 2.47154551228657390587241844980, 3.11607214735780044486224253512, 4.14960916083854517767813607157, 4.69979994823695534585258083865, 5.31674636481810275121662786152, 6.18576355541795905515003569702, 6.48609200975402783303901709164, 7.40688436583773574748703499008

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