Properties

Label 2-1805-1.1-c1-0-12
Degree $2$
Conductor $1805$
Sign $1$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
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  + 0.274·2-s − 3.09·3-s − 1.92·4-s − 5-s − 0.849·6-s + 3.63·7-s − 1.07·8-s + 6.57·9-s − 0.274·10-s + 2.45·11-s + 5.95·12-s − 3.81·13-s + 0.999·14-s + 3.09·15-s + 3.55·16-s − 3.62·17-s + 1.80·18-s + 1.92·20-s − 11.2·21-s + 0.674·22-s − 9.19·23-s + 3.33·24-s + 25-s − 1.04·26-s − 11.0·27-s − 7.00·28-s + 1.81·29-s + ⋯
L(s)  = 1  + 0.194·2-s − 1.78·3-s − 0.962·4-s − 0.447·5-s − 0.346·6-s + 1.37·7-s − 0.381·8-s + 2.19·9-s − 0.0868·10-s + 0.740·11-s + 1.71·12-s − 1.05·13-s + 0.267·14-s + 0.798·15-s + 0.888·16-s − 0.878·17-s + 0.425·18-s + 0.430·20-s − 2.45·21-s + 0.143·22-s − 1.91·23-s + 0.680·24-s + 0.200·25-s − 0.205·26-s − 2.12·27-s − 1.32·28-s + 0.337·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5957685277\)
\(L(\frac12)\) \(\approx\) \(0.5957685277\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - 0.274T + 2T^{2} \)
3 \( 1 + 3.09T + 3T^{2} \)
7 \( 1 - 3.63T + 7T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + 3.81T + 13T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
23 \( 1 + 9.19T + 23T^{2} \)
29 \( 1 - 1.81T + 29T^{2} \)
31 \( 1 + 7.04T + 31T^{2} \)
37 \( 1 - 0.500T + 37T^{2} \)
41 \( 1 - 1.42T + 41T^{2} \)
43 \( 1 - 6.78T + 43T^{2} \)
47 \( 1 - 0.193T + 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 - 1.29T + 59T^{2} \)
61 \( 1 - 8.12T + 61T^{2} \)
67 \( 1 - 8.17T + 67T^{2} \)
71 \( 1 + 5.69T + 71T^{2} \)
73 \( 1 - 2.63T + 73T^{2} \)
79 \( 1 - 5.11T + 79T^{2} \)
83 \( 1 + 0.287T + 83T^{2} \)
89 \( 1 + 4.91T + 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−9.417029429229614559602798728770, −8.407239321204324944422745810557, −7.61629493341275216371095755451, −6.78245088690356699443758449832, −5.75801946561321023382138356932, −5.21147810053026660780780406665, −4.34900122391847587230392065927, −4.10061747756962679420732962148, −1.89226926110354005974963300248, −0.56889876388102583060481163389, 0.56889876388102583060481163389, 1.89226926110354005974963300248, 4.10061747756962679420732962148, 4.34900122391847587230392065927, 5.21147810053026660780780406665, 5.75801946561321023382138356932, 6.78245088690356699443758449832, 7.61629493341275216371095755451, 8.407239321204324944422745810557, 9.417029429229614559602798728770

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