Properties

Label 2-1805-1.1-c1-0-0
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  − 1.66·2-s − 1.15·3-s + 0.770·4-s + 5-s + 1.93·6-s − 2.43·7-s + 2.04·8-s − 1.65·9-s − 1.66·10-s − 5.75·11-s − 0.893·12-s − 1.59·13-s + 4.05·14-s − 1.15·15-s − 4.94·16-s − 5.98·17-s + 2.75·18-s + 0.770·20-s + 2.82·21-s + 9.57·22-s − 0.940·23-s − 2.37·24-s + 25-s + 2.65·26-s + 5.39·27-s − 1.87·28-s + 2.61·29-s + ⋯
L(s)  = 1  − 1.17·2-s − 0.669·3-s + 0.385·4-s + 0.447·5-s + 0.788·6-s − 0.920·7-s + 0.723·8-s − 0.551·9-s − 0.526·10-s − 1.73·11-s − 0.258·12-s − 0.442·13-s + 1.08·14-s − 0.299·15-s − 1.23·16-s − 1.45·17-s + 0.649·18-s + 0.172·20-s + 0.616·21-s + 2.04·22-s − 0.196·23-s − 0.484·24-s + 0.200·25-s + 0.520·26-s + 1.03·27-s − 0.354·28-s + 0.486·29-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

\(L(1)\) \(\approx\) \(0.1526298449\)
\(L(\frac12)\) \(\approx\) \(0.1526298449\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.66T + 2T^{2} \)
3 \( 1 + 1.15T + 3T^{2} \)
7 \( 1 + 2.43T + 7T^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 + 1.59T + 13T^{2} \)
17 \( 1 + 5.98T + 17T^{2} \)
23 \( 1 + 0.940T + 23T^{2} \)
29 \( 1 - 2.61T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + 2.89T + 37T^{2} \)
41 \( 1 + 6.31T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 - 8.95T + 47T^{2} \)
53 \( 1 + 2.19T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 1.00T + 67T^{2} \)
71 \( 1 - 8.83T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 7.60T + 79T^{2} \)
83 \( 1 - 3.11T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 - 4.05T + 97T^{2} \)
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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

−9.250056929534238012382171164264, −8.672928982456712744637489878633, −7.79830742212353896298134328372, −7.00173243198180024961901028183, −6.19038164273225385999761287711, −5.32040464985557127612995396670, −4.56584503445334664662144746159, −2.97709438206567097364884155445, −2.08425907450006956228066763648, −0.30631676752128210527202595982, 0.30631676752128210527202595982, 2.08425907450006956228066763648, 2.97709438206567097364884155445, 4.56584503445334664662144746159, 5.32040464985557127612995396670, 6.19038164273225385999761287711, 7.00173243198180024961901028183, 7.79830742212353896298134328372, 8.672928982456712744637489878633, 9.250056929534238012382171164264

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