Properties

Label 2-555-185.184-c1-0-24
Degree $2$
Conductor $555$
Sign $-0.00643 + 0.999i$
Analytic cond. $4.43169$
Root an. cond. $2.10515$
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  − 1.47·2-s + i·3-s + 0.182·4-s + (−0.357 − 2.20i)5-s − 1.47i·6-s + 1.62i·7-s + 2.68·8-s − 9-s + (0.528 + 3.26i)10-s − 0.174·11-s + 0.182i·12-s − 4.30·13-s − 2.39i·14-s + (2.20 − 0.357i)15-s − 4.33·16-s + 6.78·17-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.577i·3-s + 0.0910·4-s + (−0.159 − 0.987i)5-s − 0.603i·6-s + 0.612i·7-s + 0.949·8-s − 0.333·9-s + (0.167 + 1.03i)10-s − 0.0526·11-s + 0.0525i·12-s − 1.19·13-s − 0.639i·14-s + (0.569 − 0.0923i)15-s − 1.08·16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00643 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00643 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $-0.00643 + 0.999i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ -0.00643 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315133 - 0.317169i\)
\(L(\frac12)\) \(\approx\) \(0.315133 - 0.317169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (0.357 + 2.20i)T \)
37 \( 1 + (-1.01 + 5.99i)T \)
good2 \( 1 + 1.47T + 2T^{2} \)
7 \( 1 - 1.62iT - 7T^{2} \)
11 \( 1 + 0.174T + 11T^{2} \)
13 \( 1 + 4.30T + 13T^{2} \)
17 \( 1 - 6.78T + 17T^{2} \)
19 \( 1 + 3.61iT - 19T^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 + 2.45iT - 29T^{2} \)
31 \( 1 + 6.18iT - 31T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 9.32T + 43T^{2} \)
47 \( 1 + 11.8iT - 47T^{2} \)
53 \( 1 + 5.47iT - 53T^{2} \)
59 \( 1 + 3.00iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 + 2.04iT - 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 7.16iT - 73T^{2} \)
79 \( 1 - 15.4iT - 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + 13.4iT - 89T^{2} \)
97 \( 1 + 6.23T + 97T^{2} \)
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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

−9.936019190508111248416462236105, −9.800271922243649438504182406508, −8.905354322511808285382467857659, −8.100152067546874233573923544422, −7.45882338111760017763244235619, −5.72806275513381025393800919666, −4.95696620710804462687722889636, −3.95597591454108034409120437826, −2.17890823749720021436167831361, −0.38980693100652251353173943812, 1.39986849854602011368730120269, 2.91717590164728438886521408295, 4.24687758102030480371584097950, 5.70526463038366606489783366641, 6.89472820499923511610880947401, 7.71661644840393079185137842831, 7.956905564744695965000923687106, 9.386682495780910101500117579497, 10.23147023903757749197353743230, 10.51937591269645152362725736068

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