Properties

Label 2-1805-1.1-c1-0-28
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  − 1.90·2-s − 1.90·3-s + 1.61·4-s − 5-s + 3.61·6-s − 4.23·7-s + 0.726·8-s + 0.618·9-s + 1.90·10-s − 5.85·11-s − 3.07·12-s + 3.07·13-s + 8.05·14-s + 1.90·15-s − 4.61·16-s + 5.23·17-s − 1.17·18-s − 1.61·20-s + 8.05·21-s + 11.1·22-s + 4.09·23-s − 1.38·24-s + 25-s − 5.85·26-s + 4.53·27-s − 6.85·28-s − 2.80·29-s + ⋯
L(s)  = 1  − 1.34·2-s − 1.09·3-s + 0.809·4-s − 0.447·5-s + 1.47·6-s − 1.60·7-s + 0.256·8-s + 0.206·9-s + 0.601·10-s − 1.76·11-s − 0.888·12-s + 0.853·13-s + 2.15·14-s + 0.491·15-s − 1.15·16-s + 1.26·17-s − 0.277·18-s − 0.361·20-s + 1.75·21-s + 2.37·22-s + 0.852·23-s − 0.282·24-s + 0.200·25-s − 1.14·26-s + 0.871·27-s − 1.29·28-s − 0.519·29-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 \( 1 + 1.90T + 2T^{2} \)
3 \( 1 + 1.90T + 3T^{2} \)
7 \( 1 + 4.23T + 7T^{2} \)
11 \( 1 + 5.85T + 11T^{2} \)
13 \( 1 - 3.07T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
23 \( 1 - 4.09T + 23T^{2} \)
29 \( 1 + 2.80T + 29T^{2} \)
31 \( 1 - 1.90T + 31T^{2} \)
37 \( 1 + 2.80T + 37T^{2} \)
41 \( 1 + 6.88T + 41T^{2} \)
43 \( 1 - 0.381T + 43T^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 - 3.94T + 61T^{2} \)
67 \( 1 - 5.98T + 67T^{2} \)
71 \( 1 - 0.171T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 5.25T + 79T^{2} \)
83 \( 1 + 8.76T + 83T^{2} \)
89 \( 1 - 7.77T + 89T^{2} \)
97 \( 1 + 2.62T + 97T^{2} \)
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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

−8.822893817992677938113525137451, −8.237501451223468361567529639800, −7.30512940832295648735984909465, −6.72097173069887569669087051926, −5.72283598798135121858743076746, −5.11876385316569173526160475700, −3.65461680116967875928998312304, −2.69808227559460142253821564047, −0.877302181015217455908674217002, 0, 0.877302181015217455908674217002, 2.69808227559460142253821564047, 3.65461680116967875928998312304, 5.11876385316569173526160475700, 5.72283598798135121858743076746, 6.72097173069887569669087051926, 7.30512940832295648735984909465, 8.237501451223468361567529639800, 8.822893817992677938113525137451

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