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Properties

Label 2-2205-35.34-c0-0-5
Degree $2$
Conductor $2205$
Sign $-0.912 + 0.409i$
Analytic cond. $1.10043$
Root an. cond. $1.04901$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.84i·2-s − 2.41·4-s + 5-s + 2.61i·8-s − 1.84i·10-s + 2.41·16-s + 1.41·17-s − 1.84i·19-s − 2.41·20-s + 0.765i·23-s + 25-s − 0.765i·31-s − 1.84i·32-s − 2.61i·34-s − 3.41·38-s + ⋯
L(s)  = 1  − 1.84i·2-s − 2.41·4-s + 5-s + 2.61i·8-s − 1.84i·10-s + 2.41·16-s + 1.41·17-s − 1.84i·19-s − 2.41·20-s + 0.765i·23-s + 25-s − 0.765i·31-s − 1.84i·32-s − 2.61i·34-s − 3.41·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.912 + 0.409i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.912 + 0.409i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.222425448\)
\(L(\frac12)\) \(\approx\) \(1.222425448\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good2 \( 1 + 1.84iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 + 1.84iT - T^{2} \)
23 \( 1 - 0.765iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 0.765iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + 0.765iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.765iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.330055866593967678061647183227, −8.589074342762012987982840196285, −7.56893500897618184430255706200, −6.38409892821472916916789527213, −5.30740788695277413279238695847, −4.82650543143880113316328524582, −3.62450325852020778051404174411, −2.86941877773564529767094382646, −2.02803876860897933246351956094, −1.00142626451905489443866557625, 1.45076711212592234742375456234, 3.18498130493706445505633517989, 4.28125410473772657165017981366, 5.25733543364132834790665172327, 5.76672941741185542478639628630, 6.37397819433958025698763114263, 7.16240996443883878644039329638, 8.002566253271842070096769981158, 8.493212867909974340291092695854, 9.407728851252144709700013278207

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