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Properties

Label 2-552-8.5-c1-0-34
Degree $2$
Conductor $552$
Sign $-0.887 + 0.460i$
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  + (−1.32 − 0.503i)2-s − i·3-s + (1.49 + 1.33i)4-s − 2.69i·5-s + (−0.503 + 1.32i)6-s + 1.19·7-s + (−1.30 − 2.51i)8-s − 9-s + (−1.35 + 3.56i)10-s − 3.08i·11-s + (1.33 − 1.49i)12-s + 0.0366i·13-s + (−1.58 − 0.603i)14-s − 2.69·15-s + (0.458 + 3.97i)16-s − 0.279·17-s + ⋯
L(s)  = 1  + (−0.934 − 0.355i)2-s − 0.577i·3-s + (0.746 + 0.665i)4-s − 1.20i·5-s + (−0.205 + 0.539i)6-s + 0.453·7-s + (−0.460 − 0.887i)8-s − 0.333·9-s + (−0.429 + 1.12i)10-s − 0.930i·11-s + (0.384 − 0.431i)12-s + 0.0101i·13-s + (−0.423 − 0.161i)14-s − 0.696·15-s + (0.114 + 0.993i)16-s − 0.0677·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.887 + 0.460i$
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: \((2,\ 552,\ (\ :1/2),\ -0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197001 - 0.807007i\)
\(L(\frac12)\) \(\approx\) \(0.197001 - 0.807007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.503i)T \)
3 \( 1 + iT \)
23 \( 1 - T \)
good5 \( 1 + 2.69iT - 5T^{2} \)
7 \( 1 - 1.19T + 7T^{2} \)
11 \( 1 + 3.08iT - 11T^{2} \)
13 \( 1 - 0.0366iT - 13T^{2} \)
17 \( 1 + 0.279T + 17T^{2} \)
19 \( 1 + 2.99iT - 19T^{2} \)
29 \( 1 - 1.31iT - 29T^{2} \)
31 \( 1 + 3.32T + 31T^{2} \)
37 \( 1 - 0.965iT - 37T^{2} \)
41 \( 1 - 0.122T + 41T^{2} \)
43 \( 1 + 6.45iT - 43T^{2} \)
47 \( 1 + 7.33T + 47T^{2} \)
53 \( 1 + 1.48iT - 53T^{2} \)
59 \( 1 - 0.292iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + 7.91T + 71T^{2} \)
73 \( 1 - 7.30T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 13.9iT - 83T^{2} \)
89 \( 1 - 2.48T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−10.45584491142998156973544946850, −9.205053519530954379000966228958, −8.708735108802231528666371421178, −8.017551993898380183622573985621, −7.06482815203853781687184237455, −5.92978692923479354258228158556, −4.77029126189725196508017836974, −3.27347073422958843136272727606, −1.80736316687207299713392355937, −0.64960703030249531000080936564, 1.93939371284990620725943110900, 3.18858829172446762856574259139, 4.69010710227475121771990532756, 5.88357392224115248702273487198, 6.82990281797856398144465037886, 7.59401256775689153127422962193, 8.483160854070268269956799682887, 9.647516607485069503237068580695, 10.09521235406960896134253100098, 11.00602704566823990773944815894

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