Properties

Label 2-1805-1.1-c1-0-79
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.726·3-s − 2·4-s − 5-s + 1.61·7-s − 2.47·9-s + 0.854·11-s − 1.45·12-s − 0.726·13-s − 0.726·15-s + 4·16-s + 3.85·17-s + 2·20-s + 1.17·21-s − 3.47·23-s + 25-s − 3.97·27-s − 3.23·28-s + 2.35·29-s − 6.88·31-s + 0.620·33-s − 1.61·35-s + 4.94·36-s + 7.77·37-s − 0.527·39-s − 5.98·41-s − 9.61·43-s − 1.70·44-s + ⋯
L(s)  = 1  + 0.419·3-s − 4-s − 0.447·5-s + 0.611·7-s − 0.824·9-s + 0.257·11-s − 0.419·12-s − 0.201·13-s − 0.187·15-s + 16-s + 0.934·17-s + 0.447·20-s + 0.256·21-s − 0.723·23-s + 0.200·25-s − 0.765·27-s − 0.611·28-s + 0.436·29-s − 1.23·31-s + 0.108·33-s − 0.273·35-s + 0.824·36-s + 1.27·37-s − 0.0845·39-s − 0.934·41-s − 1.46·43-s − 0.257·44-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: \(1\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T^{2} \)
3 \( 1 - 0.726T + 3T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 - 0.854T + 11T^{2} \)
13 \( 1 + 0.726T + 13T^{2} \)
17 \( 1 - 3.85T + 17T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 6.88T + 31T^{2} \)
37 \( 1 - 7.77T + 37T^{2} \)
41 \( 1 + 5.98T + 41T^{2} \)
43 \( 1 + 9.61T + 43T^{2} \)
47 \( 1 + 8.70T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 - 2.62T + 59T^{2} \)
61 \( 1 + 2.76T + 61T^{2} \)
67 \( 1 - 3.35T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 7.77T + 97T^{2} \)
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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

−8.779875993861071674021218387406, −8.060095466339856167971881853693, −7.77472084797823971888087692547, −6.42676314708237240886196993022, −5.42571300836847964032064170405, −4.77783931152790934855537850894, −3.77657602642235815302396227066, −3.07140620879446014618282527088, −1.57392083943964158900590007818, 0, 1.57392083943964158900590007818, 3.07140620879446014618282527088, 3.77657602642235815302396227066, 4.77783931152790934855537850894, 5.42571300836847964032064170405, 6.42676314708237240886196993022, 7.77472084797823971888087692547, 8.060095466339856167971881853693, 8.779875993861071674021218387406

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