Properties

Label 2-1805-1.1-c1-0-67
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  − 2.10·2-s + 2.97·3-s + 2.41·4-s + 5-s − 6.24·6-s + 4.82·7-s − 0.870·8-s + 5.82·9-s − 2.10·10-s + 2·11-s + 7.17·12-s − 1.23·13-s − 10.1·14-s + 2.97·15-s − 2.99·16-s − 3.65·17-s − 12.2·18-s + 2.41·20-s + 14.3·21-s − 4.20·22-s + 4.82·23-s − 2.58·24-s + 25-s + 2.58·26-s + 8.40·27-s + 11.6·28-s − 2.46·29-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.71·3-s + 1.20·4-s + 0.447·5-s − 2.54·6-s + 1.82·7-s − 0.307·8-s + 1.94·9-s − 0.664·10-s + 0.603·11-s + 2.07·12-s − 0.341·13-s − 2.71·14-s + 0.767·15-s − 0.749·16-s − 0.886·17-s − 2.88·18-s + 0.539·20-s + 3.13·21-s − 0.895·22-s + 1.00·23-s − 0.527·24-s + 0.200·25-s + 0.507·26-s + 1.61·27-s + 2.20·28-s − 0.457·29-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: $\chi_{1805} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.151026300\)
\(L(\frac12)\) \(\approx\) \(2.151026300\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.10T + 2T^{2} \)
3 \( 1 - 2.97T + 3T^{2} \)
7 \( 1 - 4.82T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 - 5.94T + 31T^{2} \)
37 \( 1 + 7.17T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 0.828T + 43T^{2} \)
47 \( 1 + 6.48T + 47T^{2} \)
53 \( 1 + 4.71T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + 5.43T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 0.343T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + 9.42T + 89T^{2} \)
97 \( 1 + 1.23T + 97T^{2} \)
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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

−9.026901157581824321918778958049, −8.626812646291738228037016753216, −7.999402062914303640890190080722, −7.40808951861491700738127797782, −6.64635311603530899777074428683, −4.96204739856865772775916960489, −4.25738903295857105579407428790, −2.83374550228748721832811747765, −1.89172705282043803470197557471, −1.39210179654541350437051642779, 1.39210179654541350437051642779, 1.89172705282043803470197557471, 2.83374550228748721832811747765, 4.25738903295857105579407428790, 4.96204739856865772775916960489, 6.64635311603530899777074428683, 7.40808951861491700738127797782, 7.999402062914303640890190080722, 8.626812646291738228037016753216, 9.026901157581824321918778958049

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