Properties

Label 2-1805-1.1-c1-0-50
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  − 1.17·2-s − 1.17·3-s − 0.618·4-s − 5-s + 1.38·6-s + 0.236·7-s + 3.07·8-s − 1.61·9-s + 1.17·10-s + 0.854·11-s + 0.726·12-s − 0.726·13-s − 0.277·14-s + 1.17·15-s − 2.38·16-s + 0.763·17-s + 1.90·18-s + 0.618·20-s − 0.277·21-s − 1.00·22-s − 7.09·23-s − 3.61·24-s + 25-s + 0.854·26-s + 5.42·27-s − 0.145·28-s + 8.78·29-s + ⋯
L(s)  = 1  − 0.831·2-s − 0.678·3-s − 0.309·4-s − 0.447·5-s + 0.564·6-s + 0.0892·7-s + 1.08·8-s − 0.539·9-s + 0.371·10-s + 0.257·11-s + 0.209·12-s − 0.201·13-s − 0.0741·14-s + 0.303·15-s − 0.595·16-s + 0.185·17-s + 0.448·18-s + 0.138·20-s − 0.0605·21-s − 0.214·22-s − 1.47·23-s − 0.738·24-s + 0.200·25-s + 0.167·26-s + 1.04·27-s − 0.0275·28-s + 1.63·29-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: \(1\)
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 \( 1 + 1.17T + 2T^{2} \)
3 \( 1 + 1.17T + 3T^{2} \)
7 \( 1 - 0.236T + 7T^{2} \)
11 \( 1 - 0.854T + 11T^{2} \)
13 \( 1 + 0.726T + 13T^{2} \)
17 \( 1 - 0.763T + 17T^{2} \)
23 \( 1 + 7.09T + 23T^{2} \)
29 \( 1 - 8.78T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 - 8.78T + 37T^{2} \)
41 \( 1 + 1.62T + 41T^{2} \)
43 \( 1 - 2.61T + 43T^{2} \)
47 \( 1 - 7.47T + 47T^{2} \)
53 \( 1 + 1.00T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 8.50T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 8.33T + 89T^{2} \)
97 \( 1 + 4.25T + 97T^{2} \)
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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

−8.758048705604238838057153952581, −8.222592240844928128461243450097, −7.53217882328957006895626933987, −6.51987916468680208258760091391, −5.71174689487335185254999396649, −4.72457422432589729077195513728, −4.05101507018373445083029355885, −2.67824812665464001826250491565, −1.14823418666610834223641204809, 0, 1.14823418666610834223641204809, 2.67824812665464001826250491565, 4.05101507018373445083029355885, 4.72457422432589729077195513728, 5.71174689487335185254999396649, 6.51987916468680208258760091391, 7.53217882328957006895626933987, 8.222592240844928128461243450097, 8.758048705604238838057153952581

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