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Properties

Label 2-1805-5.4-c1-0-124
Degree $2$
Conductor $1805$
Sign $0.223 + 0.974i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + (−0.5 − 2.17i)5-s − 4.35i·7-s + 3·9-s + 5·11-s + 4·16-s − 4.35i·17-s + (−1 − 4.35i)20-s + 8.71i·23-s + (−4.50 + 2.17i)25-s − 8.71i·28-s + (−9.50 + 2.17i)35-s + 6·36-s + 13.0i·43-s + 10·44-s + (−1.5 − 6.53i)45-s + ⋯
L(s)  = 1  + 4-s + (−0.223 − 0.974i)5-s − 1.64i·7-s + 9-s + 1.50·11-s + 16-s − 1.05i·17-s + (−0.223 − 0.974i)20-s + 1.81i·23-s + (−0.900 + 0.435i)25-s − 1.64i·28-s + (−1.60 + 0.368i)35-s + 36-s + 1.99i·43-s + 1.50·44-s + (−0.223 − 0.974i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.223 + 0.974i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.571649156\)
\(L(\frac12)\) \(\approx\) \(2.571649156\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 2.17i)T \)
19 \( 1 \)
good2 \( 1 - 2T^{2} \)
3 \( 1 - 3T^{2} \)
7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 4.35iT - 17T^{2} \)
23 \( 1 - 8.71iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 + 4.35iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.0iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8.71iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.436385365048427922057813788004, −8.066024701715080433411156361123, −7.32784909456880983184783824028, −7.01185268922805688902751875529, −6.04557888591264506973551174429, −4.81898104525831975047546655384, −4.07479631525240120646021361236, −3.37818609089530100132707616139, −1.52684418489301665659941999425, −1.09138253188100489468037931358, 1.67373409149660737061842670940, 2.40443777649611652640449082062, 3.41206874872959085804217333513, 4.33917776647577494876924816201, 5.75431925473642207948907103346, 6.44590779767483461174758937445, 6.77199392012558742286058165180, 7.79673289714273862146197698551, 8.654291319180622309730142326698, 9.420264481221764838487302267372

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