Properties

Label 2-8030-1.1-c1-0-171
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 − 1.82·3-s + 4-s − 5-s − 1.82·6-s + 3.00·7-s + 8-s + 0.335·9-s − 10-s − 11-s − 1.82·12-s − 4.32·13-s + 3.00·14-s + 1.82·15-s + 16-s − 2.32·17-s + 0.335·18-s − 1.98·19-s − 20-s − 5.48·21-s − 22-s + 6.29·23-s − 1.82·24-s + 25-s − 4.32·26-s + 4.86·27-s + 3.00·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.05·3-s + 0.5·4-s − 0.447·5-s − 0.745·6-s + 1.13·7-s + 0.353·8-s + 0.111·9-s − 0.316·10-s − 0.301·11-s − 0.527·12-s − 1.19·13-s + 0.802·14-s + 0.471·15-s + 0.250·16-s − 0.565·17-s + 0.0791·18-s − 0.454·19-s − 0.223·20-s − 1.19·21-s − 0.213·22-s + 1.31·23-s − 0.372·24-s + 0.200·25-s − 0.848·26-s + 0.936·27-s + 0.567·28-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 \( 1 + 1.82T + 3T^{2} \)
7 \( 1 - 3.00T + 7T^{2} \)
13 \( 1 + 4.32T + 13T^{2} \)
17 \( 1 + 2.32T + 17T^{2} \)
19 \( 1 + 1.98T + 19T^{2} \)
23 \( 1 - 6.29T + 23T^{2} \)
29 \( 1 - 3.78T + 29T^{2} \)
31 \( 1 - 2.81T + 31T^{2} \)
37 \( 1 + 0.856T + 37T^{2} \)
41 \( 1 + 5.60T + 41T^{2} \)
43 \( 1 - 8.01T + 43T^{2} \)
47 \( 1 + 6.59T + 47T^{2} \)
53 \( 1 + 9.34T + 53T^{2} \)
59 \( 1 - 7.84T + 59T^{2} \)
61 \( 1 + 2.89T + 61T^{2} \)
67 \( 1 + 9.07T + 67T^{2} \)
71 \( 1 - 4.45T + 71T^{2} \)
79 \( 1 - 5.72T + 79T^{2} \)
83 \( 1 + 5.58T + 83T^{2} \)
89 \( 1 + 6.01T + 89T^{2} \)
97 \( 1 + 7.84T + 97T^{2} \)
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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

−7.24752662215222304118943152181, −6.75675987948321455957899491523, −5.98845196558182427097148005112, −5.07594619713815166466460579099, −4.88658979496133879305159339081, −4.30497713180684647978611868625, −3.08632395654248289470448025631, −2.35096795936989614648694963527, −1.22223470276789757342223816014, 0, 1.22223470276789757342223816014, 2.35096795936989614648694963527, 3.08632395654248289470448025631, 4.30497713180684647978611868625, 4.88658979496133879305159339081, 5.07594619713815166466460579099, 5.98845196558182427097148005112, 6.75675987948321455957899491523, 7.24752662215222304118943152181

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