Properties

Label 2-1805-1.1-c1-0-95
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  + 2.61·2-s + 2.61·3-s + 4.85·4-s − 5-s + 6.85·6-s + 0.236·7-s + 7.47·8-s + 3.85·9-s − 2.61·10-s − 4.61·11-s + 12.7·12-s + 5·13-s + 0.618·14-s − 2.61·15-s + 9.85·16-s − 6·17-s + 10.0·18-s − 4.85·20-s + 0.618·21-s − 12.0·22-s − 1.61·23-s + 19.5·24-s + 25-s + 13.0·26-s + 2.23·27-s + 1.14·28-s − 1.85·29-s + ⋯
L(s)  = 1  + 1.85·2-s + 1.51·3-s + 2.42·4-s − 0.447·5-s + 2.79·6-s + 0.0892·7-s + 2.64·8-s + 1.28·9-s − 0.827·10-s − 1.39·11-s + 3.66·12-s + 1.38·13-s + 0.165·14-s − 0.675·15-s + 2.46·16-s − 1.45·17-s + 2.37·18-s − 1.08·20-s + 0.134·21-s − 2.57·22-s − 0.337·23-s + 3.99·24-s + 0.200·25-s + 2.56·26-s + 0.430·27-s + 0.216·28-s − 0.344·29-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

\(L(1)\) \(\approx\) \(8.005632407\)
\(L(\frac12)\) \(\approx\) \(8.005632407\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 2T^{2} \)
3 \( 1 - 2.61T + 3T^{2} \)
7 \( 1 - 0.236T + 7T^{2} \)
11 \( 1 + 4.61T + 11T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
23 \( 1 + 1.61T + 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
31 \( 1 - 4.14T + 31T^{2} \)
37 \( 1 + 1.85T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 5.61T + 53T^{2} \)
59 \( 1 - 0.0901T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 - 3.76T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + 6.70T + 89T^{2} \)
97 \( 1 - 6.09T + 97T^{2} \)
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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

−9.047443828937567089998162044729, −8.205402875684973851902928915518, −7.71588748756469928128942260479, −6.75852679799842559231476869128, −5.96046546573772756684310601278, −4.87588286437027067431880272499, −4.15754029474866119044372825238, −3.41215051482841317071662253214, −2.70341147120747461778621637303, −1.90447371278075048234866366222, 1.90447371278075048234866366222, 2.70341147120747461778621637303, 3.41215051482841317071662253214, 4.15754029474866119044372825238, 4.87588286437027067431880272499, 5.96046546573772756684310601278, 6.75852679799842559231476869128, 7.71588748756469928128942260479, 8.205402875684973851902928915518, 9.047443828937567089998162044729

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