Properties

Label 2-2e3-8.3-c38-0-14
Degree $2$
Conductor $8$
Sign $1$
Analytic cond. $73.1707$
Root an. cond. $8.55399$
Motivic weight $38$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.24e5·2-s + 4.01e8·3-s + 2.74e11·4-s − 2.10e14·6-s − 1.44e17·8-s − 1.18e18·9-s − 6.28e19·11-s + 1.10e20·12-s + 7.55e22·16-s − 4.30e23·17-s + 6.23e23·18-s + 3.20e24·19-s + 3.29e25·22-s − 5.79e25·24-s + 3.63e26·25-s − 1.02e27·27-s − 3.96e28·32-s − 2.52e28·33-s + 2.25e29·34-s − 3.26e29·36-s − 1.67e30·38-s − 8.34e30·41-s − 7.54e29·43-s − 1.72e31·44-s + 3.03e31·48-s + 1.29e32·49-s − 1.90e32·50-s + ⋯
L(s)  = 1  − 2-s + 0.345·3-s + 4-s − 0.345·6-s − 8-s − 0.880·9-s − 1.02·11-s + 0.345·12-s + 16-s − 1.79·17-s + 0.880·18-s + 1.61·19-s + 1.02·22-s − 0.345·24-s + 25-s − 0.650·27-s − 32-s − 0.355·33-s + 1.79·34-s − 0.880·36-s − 1.61·38-s − 1.89·41-s − 0.0694·43-s − 1.02·44-s + 0.345·48-s + 49-s − 50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $1$
Analytic conductor: \(73.1707\)
Root analytic conductor: \(8.55399\)
Motivic weight: \(38\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :19),\ 1)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(0.9557595616\)
\(L(\frac12)\) \(\approx\) \(0.9557595616\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{19} T \)
good3 \( 1 - 401778862 T + p^{38} T^{2} \)
5 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
7 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
11 \( 1 + 62820434121898204066 T + p^{38} T^{2} \)
13 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
17 \( 1 + \)\(43\!\cdots\!62\)\( T + p^{38} T^{2} \)
19 \( 1 - \)\(32\!\cdots\!86\)\( T + p^{38} T^{2} \)
23 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
29 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
31 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
37 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
41 \( 1 + \)\(83\!\cdots\!06\)\( T + p^{38} T^{2} \)
43 \( 1 + \)\(75\!\cdots\!14\)\( T + p^{38} T^{2} \)
47 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
53 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
59 \( 1 - \)\(79\!\cdots\!78\)\( T + p^{38} T^{2} \)
61 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
67 \( 1 + \)\(27\!\cdots\!02\)\( T + p^{38} T^{2} \)
71 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
73 \( 1 - \)\(12\!\cdots\!22\)\( T + p^{38} T^{2} \)
79 \( ( 1 - p^{19} T )( 1 + p^{19} T ) \)
83 \( 1 - \)\(54\!\cdots\!22\)\( T + p^{38} T^{2} \)
89 \( 1 - \)\(11\!\cdots\!06\)\( T + p^{38} T^{2} \)
97 \( 1 + \)\(20\!\cdots\!66\)\( T + p^{38} T^{2} \)
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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

−13.47524227353262008278783488171, −11.71858292827228083181890652308, −10.59243965397737344699975408883, −9.160242986209513506976465101791, −8.181694470107542906511450175519, −6.88088236187922687119156161451, −5.32164581086886974725893267283, −3.13348545554262342282045911132, −2.16344320293592441164032753381, −0.54979876145787308837673150238, 0.54979876145787308837673150238, 2.16344320293592441164032753381, 3.13348545554262342282045911132, 5.32164581086886974725893267283, 6.88088236187922687119156161451, 8.181694470107542906511450175519, 9.160242986209513506976465101791, 10.59243965397737344699975408883, 11.71858292827228083181890652308, 13.47524227353262008278783488171

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