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Properties

Label 2-1805-5.4-c1-0-104
Degree $2$
Conductor $1805$
Sign $0.983 - 0.182i$
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  + 2.45i·2-s − 1.56i·3-s − 4.03·4-s + (2.19 − 0.407i)5-s + 3.83·6-s − 4.50i·7-s − 4.99i·8-s + 0.563·9-s + (1 + 5.40i)10-s + 2.19·11-s + 6.29i·12-s + 3.75i·13-s + 11.0·14-s + (−0.635 − 3.43i)15-s + 4.19·16-s − 0.665i·17-s + ⋯
L(s)  = 1  + 1.73i·2-s − 0.901i·3-s − 2.01·4-s + (0.983 − 0.182i)5-s + 1.56·6-s − 1.70i·7-s − 1.76i·8-s + 0.187·9-s + (0.316 + 1.70i)10-s + 0.662·11-s + 1.81i·12-s + 1.04i·13-s + 2.95·14-s + (−0.164 − 0.886i)15-s + 1.04·16-s − 0.161i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.983 - 0.182i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.983 - 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.915188026\)
\(L(\frac12)\) \(\approx\) \(1.915188026\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.19 + 0.407i)T \)
19 \( 1 \)
good2 \( 1 - 2.45iT - 2T^{2} \)
3 \( 1 + 1.56iT - 3T^{2} \)
7 \( 1 + 4.50iT - 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 - 3.75iT - 13T^{2} \)
17 \( 1 + 0.665iT - 17T^{2} \)
23 \( 1 + 0.488iT - 23T^{2} \)
29 \( 1 - 3.59T + 29T^{2} \)
31 \( 1 + 6.83T + 31T^{2} \)
37 \( 1 + 3.01iT - 37T^{2} \)
41 \( 1 + 0.0724T + 41T^{2} \)
43 \( 1 - 0.420iT - 43T^{2} \)
47 \( 1 + 5.02iT - 47T^{2} \)
53 \( 1 + 2.61iT - 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 7.06T + 61T^{2} \)
67 \( 1 - 5.72iT - 67T^{2} \)
71 \( 1 + 6.97T + 71T^{2} \)
73 \( 1 + 2.95iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 + 1.33T + 89T^{2} \)
97 \( 1 - 4.38iT - 97T^{2} \)
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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

−9.072696031102578439904412294087, −8.226562704834241679496291743843, −7.33125393248449941525386676209, −6.85372086937448578423654893815, −6.55661901229278968232194357167, −5.55634679295280013370611392648, −4.54881555693825348280642985007, −3.90359198167529014938342011816, −1.87402131640564739846297536130, −0.790857048317052731223698787786, 1.38747949727907257300311457100, 2.37191097885002066476000687572, 3.07548626180086836327383819261, 3.97324313904158801279409584816, 5.11460759930512571528292605184, 5.51328142126264668956935047679, 6.61828032854626065475566869736, 8.254762022784322361458261072708, 9.101750944565652734459211073164, 9.382594273959836938413709938277

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