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Properties

Label 2-2205-315.88-c0-0-1
Degree $2$
Conductor $2205$
Sign $0.129 + 0.991i$
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 − i)2-s + (0.965 − 0.258i)3-s − i·4-s + (−0.965 + 0.258i)5-s + (0.707 − 1.22i)6-s + (0.866 − 0.499i)9-s + (−0.707 + 1.22i)10-s + (0.5 + 0.866i)11-s + (−0.258 − 0.965i)12-s + (0.258 − 0.965i)13-s + (−0.866 + 0.499i)15-s + 16-s + (−0.258 − 0.965i)17-s + (0.366 − 1.36i)18-s + (0.258 + 0.965i)20-s + ⋯
L(s)  = 1  + (1 − i)2-s + (0.965 − 0.258i)3-s − i·4-s + (−0.965 + 0.258i)5-s + (0.707 − 1.22i)6-s + (0.866 − 0.499i)9-s + (−0.707 + 1.22i)10-s + (0.5 + 0.866i)11-s + (−0.258 − 0.965i)12-s + (0.258 − 0.965i)13-s + (−0.866 + 0.499i)15-s + 16-s + (−0.258 − 0.965i)17-s + (0.366 − 1.36i)18-s + (0.258 + 0.965i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.129 + 0.991i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1978, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ 0.129 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.488667634\)
\(L(\frac12)\) \(\approx\) \(2.488667634\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 \)
good2 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.101116739172991541586938282661, −8.186401116994747327157240547217, −7.57988604597319614810757235621, −6.90520793660191195163877503120, −5.68162786007941199567754158985, −4.50800022155361154879620900077, −4.06317767541169927740479785348, −3.19505113570605752854307477054, −2.57327702727529903341539292137, −1.42675462840560831651623504443, 1.71145789624963292235389507376, 3.29431797246773881332284182172, 4.00080652049434116548653428667, 4.30588093731443298506063710605, 5.41074537292961454590706986330, 6.37582656521964555994609931062, 7.02265114453524293877538244755, 7.83987239969370418886985993056, 8.509253332189983391203834981826, 8.945264020236591985718499276192

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