Properties

Label 2-1805-1.1-c1-0-44
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  − 1.13·2-s − 1.70·3-s − 0.703·4-s + 5-s + 1.93·6-s − 4.75·7-s + 3.07·8-s − 0.0975·9-s − 1.13·10-s + 3.46·11-s + 1.19·12-s + 1.17·13-s + 5.41·14-s − 1.70·15-s − 2.09·16-s − 6.20·17-s + 0.111·18-s − 0.703·20-s + 8.10·21-s − 3.93·22-s + 5.05·23-s − 5.24·24-s + 25-s − 1.33·26-s + 5.27·27-s + 3.34·28-s − 1.61·29-s + ⋯
L(s)  = 1  − 0.805·2-s − 0.983·3-s − 0.351·4-s + 0.447·5-s + 0.791·6-s − 1.79·7-s + 1.08·8-s − 0.0325·9-s − 0.360·10-s + 1.04·11-s + 0.346·12-s + 0.324·13-s + 1.44·14-s − 0.439·15-s − 0.524·16-s − 1.50·17-s + 0.0261·18-s − 0.157·20-s + 1.76·21-s − 0.839·22-s + 1.05·23-s − 1.07·24-s + 0.200·25-s − 0.261·26-s + 1.01·27-s + 0.632·28-s − 0.299·29-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 \( 1 + 1.13T + 2T^{2} \)
3 \( 1 + 1.70T + 3T^{2} \)
7 \( 1 + 4.75T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 + 6.20T + 17T^{2} \)
23 \( 1 - 5.05T + 23T^{2} \)
29 \( 1 + 1.61T + 29T^{2} \)
31 \( 1 - 7.49T + 31T^{2} \)
37 \( 1 - 5.98T + 37T^{2} \)
41 \( 1 - 5.43T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 8.06T + 47T^{2} \)
53 \( 1 - 6.68T + 53T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 - 6.20T + 61T^{2} \)
67 \( 1 - 5.62T + 67T^{2} \)
71 \( 1 + 2.72T + 71T^{2} \)
73 \( 1 - 3.15T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 8.24T + 83T^{2} \)
89 \( 1 + 8.83T + 89T^{2} \)
97 \( 1 + 0.707T + 97T^{2} \)
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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

−9.008413290271915467619781608174, −8.457953187074868610546336597146, −6.94113184727374495769991084692, −6.59223318398197395373562962525, −5.90800583887361252115700949291, −4.83143647020547364163938877286, −3.92468130988421657728969581405, −2.72768537561347978172728469928, −1.08720259256496384442333576610, 0, 1.08720259256496384442333576610, 2.72768537561347978172728469928, 3.92468130988421657728969581405, 4.83143647020547364163938877286, 5.90800583887361252115700949291, 6.59223318398197395373562962525, 6.94113184727374495769991084692, 8.457953187074868610546336597146, 9.008413290271915467619781608174

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