Properties

Label 2-8041-1.1-c1-0-262
Degree $2$
Conductor $8041$
Sign $1$
Analytic cond. $64.2077$
Root an. cond. $8.01297$
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  − 1.70·2-s + 3.00·3-s + 0.914·4-s + 2.58·5-s − 5.12·6-s − 2.99·7-s + 1.85·8-s + 6.02·9-s − 4.40·10-s + 11-s + 2.74·12-s + 2.82·13-s + 5.11·14-s + 7.75·15-s − 4.99·16-s − 17-s − 10.2·18-s − 4.52·19-s + 2.36·20-s − 8.99·21-s − 1.70·22-s + 8.03·23-s + 5.56·24-s + 1.66·25-s − 4.81·26-s + 9.08·27-s − 2.73·28-s + ⋯
L(s)  = 1  − 1.20·2-s + 1.73·3-s + 0.457·4-s + 1.15·5-s − 2.09·6-s − 1.13·7-s + 0.655·8-s + 2.00·9-s − 1.39·10-s + 0.301·11-s + 0.792·12-s + 0.782·13-s + 1.36·14-s + 2.00·15-s − 1.24·16-s − 0.242·17-s − 2.42·18-s − 1.03·19-s + 0.527·20-s − 1.96·21-s − 0.363·22-s + 1.67·23-s + 1.13·24-s + 0.333·25-s − 0.944·26-s + 1.74·27-s − 0.517·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8041\)    =    \(11 \cdot 17 \cdot 43\)
Sign: $1$
Analytic conductor: \(64.2077\)
Root analytic conductor: \(8.01297\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8041,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.453201856\)
\(L(\frac12)\) \(\approx\) \(2.453201856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
17 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 + 1.70T + 2T^{2} \)
3 \( 1 - 3.00T + 3T^{2} \)
5 \( 1 - 2.58T + 5T^{2} \)
7 \( 1 + 2.99T + 7T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
19 \( 1 + 4.52T + 19T^{2} \)
23 \( 1 - 8.03T + 23T^{2} \)
29 \( 1 + 1.93T + 29T^{2} \)
31 \( 1 + 4.01T + 31T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
47 \( 1 - 5.66T + 47T^{2} \)
53 \( 1 + 3.29T + 53T^{2} \)
59 \( 1 - 3.03T + 59T^{2} \)
61 \( 1 - 7.35T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 1.74T + 71T^{2} \)
73 \( 1 - 9.72T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 0.743T + 83T^{2} \)
89 \( 1 - 2.14T + 89T^{2} \)
97 \( 1 - 7.44T + 97T^{2} \)
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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

−8.187789202328196787783461741205, −7.25798063038710659687248962631, −6.76470061855606270533246678043, −6.10878646149303074984238734919, −4.95930052179510013283776181413, −3.93039637762694812921034765437, −3.33183098026605657750096708688, −2.36636993222826690399338616588, −1.88224319253502857094130780630, −0.876519867817626614783528150571, 0.876519867817626614783528150571, 1.88224319253502857094130780630, 2.36636993222826690399338616588, 3.33183098026605657750096708688, 3.93039637762694812921034765437, 4.95930052179510013283776181413, 6.10878646149303074984238734919, 6.76470061855606270533246678043, 7.25798063038710659687248962631, 8.187789202328196787783461741205

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