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Properties

Label 2-2205-5.4-c1-0-87
Degree $2$
Conductor $2205$
Sign $i$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 2.23i·5-s + 3·11-s − 6.70i·13-s + 4·16-s − 2.23i·17-s − 4.47i·20-s − 5.00·25-s − 9·29-s + 6·44-s + 11.1i·47-s − 13.4i·52-s − 6.70i·55-s + 8·64-s − 15.0·65-s + ⋯
L(s)  = 1  + 4-s − 0.999i·5-s + 0.904·11-s − 1.86i·13-s + 16-s − 0.542i·17-s − 0.999i·20-s − 1.00·25-s − 1.67·29-s + 0.904·44-s + 1.63i·47-s − 1.86i·52-s − 0.904i·55-s + 64-s − 1.86·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $i$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.255568197\)
\(L(\frac12)\) \(\approx\) \(2.255568197\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
7 \( 1 \)
good2 \( 1 - 2T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 6.70iT - 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 6.70iT - 97T^{2} \)
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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

−8.894906836104855504452077993308, −7.85151571939304969434131569200, −7.56975380788859788257358982809, −6.40891496103094300770859251850, −5.71675920082583197683234924659, −5.03260973859624312324749565359, −3.84103101574617743385933003172, −2.98832972993189075799072485191, −1.79660858322491714909986436321, −0.75247110204299960144919557136, 1.66710869575317664509841209561, 2.27649644787652347374740204513, 3.54834642237086968305671203142, 4.07497607912174528141005543020, 5.54097090215634869573652336963, 6.40204357860456518017262622576, 6.84833226466404512755412142014, 7.39471658529375921098455680408, 8.436855195587168631105189545732, 9.371059841556066167526026663404

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