Properties

Label 2-8030-1.1-c1-0-53
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.53·3-s + 4-s + 5-s + 2.53·6-s − 3.43·7-s − 8-s + 3.42·9-s − 10-s − 11-s − 2.53·12-s + 3.03·13-s + 3.43·14-s − 2.53·15-s + 16-s + 6.15·17-s − 3.42·18-s + 1.67·19-s + 20-s + 8.70·21-s + 22-s + 6.66·23-s + 2.53·24-s + 25-s − 3.03·26-s − 1.07·27-s − 3.43·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.46·3-s + 0.5·4-s + 0.447·5-s + 1.03·6-s − 1.29·7-s − 0.353·8-s + 1.14·9-s − 0.316·10-s − 0.301·11-s − 0.731·12-s + 0.841·13-s + 0.918·14-s − 0.654·15-s + 0.250·16-s + 1.49·17-s − 0.806·18-s + 0.383·19-s + 0.223·20-s + 1.90·21-s + 0.213·22-s + 1.38·23-s + 0.517·24-s + 0.200·25-s − 0.595·26-s − 0.206·27-s − 0.649·28-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: \(0\)
Selberg data: \((2,\ 8030,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8796034209\)
\(L(\frac12)\) \(\approx\) \(0.8796034209\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.53T + 3T^{2} \)
7 \( 1 + 3.43T + 7T^{2} \)
13 \( 1 - 3.03T + 13T^{2} \)
17 \( 1 - 6.15T + 17T^{2} \)
19 \( 1 - 1.67T + 19T^{2} \)
23 \( 1 - 6.66T + 23T^{2} \)
29 \( 1 - 4.83T + 29T^{2} \)
31 \( 1 + 7.81T + 31T^{2} \)
37 \( 1 - 6.51T + 37T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 - 1.00T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 3.90T + 59T^{2} \)
61 \( 1 + 8.61T + 61T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 - 0.852T + 71T^{2} \)
79 \( 1 + 8.44T + 79T^{2} \)
83 \( 1 - 5.19T + 83T^{2} \)
89 \( 1 + 9.00T + 89T^{2} \)
97 \( 1 + 8.64T + 97T^{2} \)
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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

−7.55700677046303632326669146637, −7.11609209088772845876275350399, −6.35355560640305175779449193109, −5.71417963071241053874959411451, −5.58216575991801977095152668882, −4.40132611606412944014507061824, −3.34411714161963739749155142487, −2.68677157333622409003341022187, −1.20417843359550858469915788248, −0.66542786555126750746015818165, 0.66542786555126750746015818165, 1.20417843359550858469915788248, 2.68677157333622409003341022187, 3.34411714161963739749155142487, 4.40132611606412944014507061824, 5.58216575991801977095152668882, 5.71417963071241053874959411451, 6.35355560640305175779449193109, 7.11609209088772845876275350399, 7.55700677046303632326669146637

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