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Properties

Label 2-552-8.5-c1-0-25
Degree $2$
Conductor $552$
Sign $-0.285 + 0.958i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
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  + (−0.585 − 1.28i)2-s + i·3-s + (−1.31 + 1.50i)4-s − 1.66i·5-s + (1.28 − 0.585i)6-s − 0.540·7-s + (2.71 + 0.808i)8-s − 9-s + (−2.14 + 0.974i)10-s + 3.06i·11-s + (−1.50 − 1.31i)12-s − 6.59i·13-s + (0.316 + 0.696i)14-s + 1.66·15-s + (−0.545 − 3.96i)16-s + 2.00·17-s + ⋯
L(s)  = 1  + (−0.414 − 0.910i)2-s + 0.577i·3-s + (−0.657 + 0.753i)4-s − 0.743i·5-s + (0.525 − 0.239i)6-s − 0.204·7-s + (0.958 + 0.285i)8-s − 0.333·9-s + (−0.677 + 0.308i)10-s + 0.923i·11-s + (−0.435 − 0.379i)12-s − 1.82i·13-s + (0.0846 + 0.186i)14-s + 0.429·15-s + (−0.136 − 0.990i)16-s + 0.486·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.285 + 0.958i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.285 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562803 - 0.755298i\)
\(L(\frac12)\) \(\approx\) \(0.562803 - 0.755298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.585 + 1.28i)T \)
3 \( 1 - iT \)
23 \( 1 - T \)
good5 \( 1 + 1.66iT - 5T^{2} \)
7 \( 1 + 0.540T + 7T^{2} \)
11 \( 1 - 3.06iT - 11T^{2} \)
13 \( 1 + 6.59iT - 13T^{2} \)
17 \( 1 - 2.00T + 17T^{2} \)
19 \( 1 + 4.41iT - 19T^{2} \)
29 \( 1 + 3.35iT - 29T^{2} \)
31 \( 1 - 1.79T + 31T^{2} \)
37 \( 1 + 7.72iT - 37T^{2} \)
41 \( 1 + 0.419T + 41T^{2} \)
43 \( 1 + 9.98iT - 43T^{2} \)
47 \( 1 - 2.45T + 47T^{2} \)
53 \( 1 + 7.38iT - 53T^{2} \)
59 \( 1 + 5.51iT - 59T^{2} \)
61 \( 1 - 9.35iT - 61T^{2} \)
67 \( 1 - 4.53iT - 67T^{2} \)
71 \( 1 - 6.86T + 71T^{2} \)
73 \( 1 + 4.18T + 73T^{2} \)
79 \( 1 + 8.69T + 79T^{2} \)
83 \( 1 - 18.0iT - 83T^{2} \)
89 \( 1 - 4.04T + 89T^{2} \)
97 \( 1 - 5.61T + 97T^{2} \)
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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

−10.37678642946534198892157265602, −9.807359791507692764278197083128, −8.953938323744188794082594290889, −8.182092396387927039481805546500, −7.22509385324066724220126062911, −5.45960394962423683190931680562, −4.75011500775630127897163742007, −3.61307070676019297611275308320, −2.49039812872289211082915656667, −0.68062263630425815414286876627, 1.48735764764502454964163915815, 3.24571221922204437347361939724, 4.64617509794004665793479574692, 6.03145774118347318039261559366, 6.52253172315570957116078383519, 7.35083829016036061358061391870, 8.277938387389137747081601495107, 9.098523432794181261905515190543, 10.01081392566257182381499717285, 10.97273244974133701309717956641

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