Properties

Label 2-5819-1.1-c1-0-11
Degree $2$
Conductor $5819$
Sign $1$
Analytic cond. $46.4649$
Root an. cond. $6.81652$
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  − 1.66·2-s − 1.56·3-s + 0.760·4-s − 0.618·5-s + 2.60·6-s + 1.60·7-s + 2.05·8-s − 0.540·9-s + 1.02·10-s + 11-s − 1.19·12-s − 4.28·13-s − 2.67·14-s + 0.970·15-s − 4.94·16-s − 2.36·17-s + 0.898·18-s − 7.82·19-s − 0.470·20-s − 2.52·21-s − 1.66·22-s − 3.22·24-s − 4.61·25-s + 7.11·26-s + 5.55·27-s + 1.22·28-s − 9.55·29-s + ⋯
L(s)  = 1  − 1.17·2-s − 0.905·3-s + 0.380·4-s − 0.276·5-s + 1.06·6-s + 0.608·7-s + 0.728·8-s − 0.180·9-s + 0.325·10-s + 0.301·11-s − 0.344·12-s − 1.18·13-s − 0.714·14-s + 0.250·15-s − 1.23·16-s − 0.574·17-s + 0.211·18-s − 1.79·19-s − 0.105·20-s − 0.550·21-s − 0.354·22-s − 0.659·24-s − 0.923·25-s + 1.39·26-s + 1.06·27-s + 0.231·28-s − 1.77·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5819\)    =    \(11 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(46.4649\)
Root analytic conductor: \(6.81652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5819,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05308435163\)
\(L(\frac12)\) \(\approx\) \(0.05308435163\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
23 \( 1 \)
good2 \( 1 + 1.66T + 2T^{2} \)
3 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 + 0.618T + 5T^{2} \)
7 \( 1 - 1.60T + 7T^{2} \)
13 \( 1 + 4.28T + 13T^{2} \)
17 \( 1 + 2.36T + 17T^{2} \)
19 \( 1 + 7.82T + 19T^{2} \)
29 \( 1 + 9.55T + 29T^{2} \)
31 \( 1 + 6.15T + 31T^{2} \)
37 \( 1 + 8.68T + 37T^{2} \)
41 \( 1 + 2.31T + 41T^{2} \)
43 \( 1 - 4.99T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 0.722T + 53T^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 + 3.55T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 1.36T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 7.06T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 3.97T + 89T^{2} \)
97 \( 1 - 3.17T + 97T^{2} \)
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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

−8.179617850476327272918017306513, −7.43073353129963776086029974008, −6.97010058956489748366377195003, −6.01961991339931643217387370611, −5.27801404582693633870599398133, −4.53730556330152525270862469018, −3.87981181461340546740423077023, −2.30858745797831113361236743584, −1.66137668027076078250751606765, −0.15010173277645777575251467316, 0.15010173277645777575251467316, 1.66137668027076078250751606765, 2.30858745797831113361236743584, 3.87981181461340546740423077023, 4.53730556330152525270862469018, 5.27801404582693633870599398133, 6.01961991339931643217387370611, 6.97010058956489748366377195003, 7.43073353129963776086029974008, 8.179617850476327272918017306513

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