Properties

Label 2-8034-1.1-c1-0-99
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 + 1.30·5-s − 6-s + 2.83·7-s − 8-s + 9-s − 1.30·10-s + 5.04·11-s + 12-s − 13-s − 2.83·14-s + 1.30·15-s + 16-s + 2.94·17-s − 18-s + 3.34·19-s + 1.30·20-s + 2.83·21-s − 5.04·22-s − 0.726·23-s − 24-s − 3.28·25-s + 26-s + 27-s + 2.83·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.585·5-s − 0.408·6-s + 1.07·7-s − 0.353·8-s + 0.333·9-s − 0.413·10-s + 1.51·11-s + 0.288·12-s − 0.277·13-s − 0.758·14-s + 0.337·15-s + 0.250·16-s + 0.714·17-s − 0.235·18-s + 0.767·19-s + 0.292·20-s + 0.619·21-s − 1.07·22-s − 0.151·23-s − 0.204·24-s − 0.657·25-s + 0.196·26-s + 0.192·27-s + 0.536·28-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

\(L(1)\) \(\approx\) \(2.916255303\)
\(L(\frac12)\) \(\approx\) \(2.916255303\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.30T + 5T^{2} \)
7 \( 1 - 2.83T + 7T^{2} \)
11 \( 1 - 5.04T + 11T^{2} \)
17 \( 1 - 2.94T + 17T^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 + 0.726T + 23T^{2} \)
29 \( 1 + 5.62T + 29T^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 - 2.36T + 41T^{2} \)
43 \( 1 - 4.27T + 43T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 9.82T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 3.16T + 67T^{2} \)
71 \( 1 - 5.90T + 71T^{2} \)
73 \( 1 + 8.75T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 9.61T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 4.99T + 97T^{2} \)
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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

−7.952523088044615061404477253429, −7.28866423530966529131074689450, −6.66851784134939256446743338561, −5.75788380676650863240398295911, −5.18686197840959254437060639032, −4.09363961185404648443446337728, −3.51063885237823249621083382518, −2.34750541320135605085947071667, −1.69825698498526284559393444895, −1.00289729427175656740876183143, 1.00289729427175656740876183143, 1.69825698498526284559393444895, 2.34750541320135605085947071667, 3.51063885237823249621083382518, 4.09363961185404648443446337728, 5.18686197840959254437060639032, 5.75788380676650863240398295911, 6.66851784134939256446743338561, 7.28866423530966529131074689450, 7.952523088044615061404477253429

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