Properties

Label 2-2205-315.247-c0-0-1
Degree $2$
Conductor $2205$
Sign $0.786 - 0.617i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ 0.786 - 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.835424234\)
\(L(\frac12)\) \(\approx\) \(1.835424234\)
\(L(1)\) 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

−9.381601251395253617841443522048, −8.198176206532694131299684868095, −7.40493194823298204128589871952, −6.64674208455232083729941800512, −6.09477128201828613346567756843, −5.50136808224775155380848334699, −4.96960490123128407764820297205, −3.90233052221617673249760603785, −2.76113752322985671704048455525, −1.22069793938991758964957970321, 1.64664842308658229977679536253, 2.05154461334820391623772397474, 3.70699514942625116558255910889, 4.22922172053155203452116963477, 5.10231697734268096186051792264, 5.68698124698926929151473915262, 6.49214448047556063450500917969, 7.30773496840162684971847667364, 8.661235633436585184371697521026, 9.556099229942269262814526423913

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