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Properties

Label 2-2205-315.193-c0-0-1
Degree $2$
Conductor $2205$
Sign $-0.460 - 0.887i$
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  + (0.366 + 1.36i)2-s + (0.965 − 0.258i)3-s + (−0.866 + 0.5i)4-s + (0.707 + 0.707i)5-s + (0.707 + 1.22i)6-s + (0.866 − 0.499i)9-s + (−0.707 + 1.22i)10-s − 11-s + (−0.707 + 0.707i)12-s + (−0.965 + 0.258i)13-s + (0.866 + 0.500i)15-s + (−0.499 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (0.999 + i)18-s + (−0.965 − 0.258i)20-s + ⋯
L(s)  = 1  + (0.366 + 1.36i)2-s + (0.965 − 0.258i)3-s + (−0.866 + 0.5i)4-s + (0.707 + 0.707i)5-s + (0.707 + 1.22i)6-s + (0.866 − 0.499i)9-s + (−0.707 + 1.22i)10-s − 11-s + (−0.707 + 0.707i)12-s + (−0.965 + 0.258i)13-s + (0.866 + 0.500i)15-s + (−0.499 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (0.999 + i)18-s + (−0.965 − 0.258i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.460 - 0.887i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.125970862\)
\(L(\frac12)\) \(\approx\) \(2.125970862\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 \)
good2 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.242102302757960696663104489968, −8.560946405686380774403749024313, −7.57657254061851997165491469731, −7.12257274506345098553931515993, −6.76233864811707308397697579919, −5.44900917355980468894363945396, −5.13708895803513401018709210494, −3.85340148620336552800488728216, −2.74047887863719075596416519204, −2.02883375855315894041443445438, 1.31798782099869712994891835577, 2.40623551335910583946883101329, 2.77755623099169892667870167539, 3.96540446956083326741142694378, 4.75186543820213063876854442896, 5.30940041145727258328490189066, 6.70113518916332858186216721432, 7.69435189429757325836637938816, 8.495346239122957141339942565651, 9.176907067543442049023468080801

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