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Properties

Label 2-2205-315.142-c0-0-0
Degree $2$
Conductor $2205$
Sign $-0.950 + 0.311i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1402, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8037744859\)
\(L(\frac12)\) \(\approx\) \(0.8037744859\)
\(L(1)\) 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

−9.110185336738951971653947930552, −8.069827294207415547810143501969, −7.11705154723777926788601284049, −6.70645287523933360354392465448, −5.42451170941214246030439854197, −4.76325779911975879511345504432, −3.81883210011701740015901446355, −3.00847729282185499126667754578, −2.03249361882855522558216657769, −0.61568787878961620416984885558, 1.36684400846266753252146556767, 3.48958808834540493914825623444, 4.27397493926364086436225471642, 5.22664779007291524213158345624, 5.47301331389938672881475564979, 6.37747183203105333072164282662, 7.17642249963001028234461959597, 7.900390184389247460041935387438, 8.479407766467054957243202882740, 9.346685760504907960665439278677

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