Properties

Label 2-1805-1.1-c1-0-42
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  − 1.69·2-s + 0.918·3-s + 0.871·4-s − 5-s − 1.55·6-s + 3.12·7-s + 1.91·8-s − 2.15·9-s + 1.69·10-s + 5.68·11-s + 0.799·12-s + 5.35·13-s − 5.29·14-s − 0.918·15-s − 4.98·16-s + 7.27·17-s + 3.65·18-s − 0.871·20-s + 2.87·21-s − 9.63·22-s + 1.74·23-s + 1.75·24-s + 25-s − 9.07·26-s − 4.73·27-s + 2.72·28-s − 7.78·29-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.530·3-s + 0.435·4-s − 0.447·5-s − 0.635·6-s + 1.18·7-s + 0.676·8-s − 0.718·9-s + 0.535·10-s + 1.71·11-s + 0.230·12-s + 1.48·13-s − 1.41·14-s − 0.237·15-s − 1.24·16-s + 1.76·17-s + 0.861·18-s − 0.194·20-s + 0.626·21-s − 2.05·22-s + 0.364·23-s + 0.358·24-s + 0.200·25-s − 1.77·26-s − 0.911·27-s + 0.514·28-s − 1.44·29-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

\(L(1)\) \(\approx\) \(1.345381659\)
\(L(\frac12)\) \(\approx\) \(1.345381659\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69T + 2T^{2} \)
3 \( 1 - 0.918T + 3T^{2} \)
7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 - 5.68T + 11T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 - 7.27T + 17T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 + 7.78T + 29T^{2} \)
31 \( 1 - 2.11T + 31T^{2} \)
37 \( 1 + 4.72T + 37T^{2} \)
41 \( 1 + 0.302T + 41T^{2} \)
43 \( 1 - 3.54T + 43T^{2} \)
47 \( 1 + 0.288T + 47T^{2} \)
53 \( 1 + 9.14T + 53T^{2} \)
59 \( 1 - 0.589T + 59T^{2} \)
61 \( 1 - 7.80T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 1.03T + 73T^{2} \)
79 \( 1 + 5.70T + 79T^{2} \)
83 \( 1 + 3.71T + 83T^{2} \)
89 \( 1 - 5.54T + 89T^{2} \)
97 \( 1 - 6.81T + 97T^{2} \)
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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

−8.982805644319774695627543301096, −8.608642801989170336708915696832, −7.954678862142801944762588322054, −7.37412786139010721776163900337, −6.24421784132738128170708222607, −5.24952130363674718045529642399, −4.04075382327295068699718893255, −3.41216879442282087293091122533, −1.72356786553211435473065316434, −1.05485392267289955809435740553, 1.05485392267289955809435740553, 1.72356786553211435473065316434, 3.41216879442282087293091122533, 4.04075382327295068699718893255, 5.24952130363674718045529642399, 6.24421784132738128170708222607, 7.37412786139010721776163900337, 7.954678862142801944762588322054, 8.608642801989170336708915696832, 8.982805644319774695627543301096

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