Properties

Label 2-8034-1.1-c1-0-49
Degree $2$
Conductor $8034$
Sign $1$
Analytic cond. $64.1518$
Root an. cond. $8.00948$
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 − 3-s + 4-s − 0.714·5-s − 6-s + 1.94·7-s + 8-s + 9-s − 0.714·10-s − 2.05·11-s − 12-s + 13-s + 1.94·14-s + 0.714·15-s + 16-s − 7.24·17-s + 18-s + 3.15·19-s − 0.714·20-s − 1.94·21-s − 2.05·22-s + 2.78·23-s − 24-s − 4.48·25-s + 26-s − 27-s + 1.94·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.319·5-s − 0.408·6-s + 0.734·7-s + 0.353·8-s + 0.333·9-s − 0.225·10-s − 0.618·11-s − 0.288·12-s + 0.277·13-s + 0.519·14-s + 0.184·15-s + 0.250·16-s − 1.75·17-s + 0.235·18-s + 0.723·19-s − 0.159·20-s − 0.424·21-s − 0.437·22-s + 0.581·23-s − 0.204·24-s − 0.897·25-s + 0.196·26-s − 0.192·27-s + 0.367·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $1$
Analytic conductor: \(64.1518\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.475794332\)
\(L(\frac12)\) \(\approx\) \(2.475794332\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 - T \)
good5 \( 1 + 0.714T + 5T^{2} \)
7 \( 1 - 1.94T + 7T^{2} \)
11 \( 1 + 2.05T + 11T^{2} \)
17 \( 1 + 7.24T + 17T^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 - 3.55T + 29T^{2} \)
31 \( 1 - 3.15T + 31T^{2} \)
37 \( 1 + 6.57T + 37T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 - 5.71T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + 9.25T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 3.69T + 67T^{2} \)
71 \( 1 - 9.88T + 71T^{2} \)
73 \( 1 - 0.674T + 73T^{2} \)
79 \( 1 + 2.88T + 79T^{2} \)
83 \( 1 - 7.36T + 83T^{2} \)
89 \( 1 - 2.45T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−7.73486411488295325089686752372, −6.95569079233755459974873005947, −6.47580168297787108769672567135, −5.52495794326033552354842306381, −5.07341066998293257264293973467, −4.37995032433286431698614376393, −3.75124618500883148556111215544, −2.66090905839876371595334306645, −1.90854106772840480328941850875, −0.71533360484807405265475362636, 0.71533360484807405265475362636, 1.90854106772840480328941850875, 2.66090905839876371595334306645, 3.75124618500883148556111215544, 4.37995032433286431698614376393, 5.07341066998293257264293973467, 5.52495794326033552354842306381, 6.47580168297787108769672567135, 6.95569079233755459974873005947, 7.73486411488295325089686752372

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