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Properties

Label 2-504-168.11-c1-0-28
Degree $2$
Conductor $504$
Sign $-0.714 + 0.699i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (1.07 − 0.921i)2-s + (0.300 − 1.97i)4-s + (−0.316 − 0.548i)5-s + (−2.06 − 1.65i)7-s + (−1.50 − 2.39i)8-s + (−0.846 − 0.296i)10-s + (0.424 + 0.245i)11-s − 3.13i·13-s + (−3.73 + 0.125i)14-s + (−3.81 − 1.18i)16-s + (−0.987 − 0.569i)17-s + (−0.591 − 1.02i)19-s + (−1.18 + 0.462i)20-s + (0.681 − 0.128i)22-s + (2.80 + 4.85i)23-s + ⋯
L(s)  = 1  + (0.758 − 0.651i)2-s + (0.150 − 0.988i)4-s + (−0.141 − 0.245i)5-s + (−0.779 − 0.626i)7-s + (−0.530 − 0.847i)8-s + (−0.267 − 0.0937i)10-s + (0.127 + 0.0738i)11-s − 0.868i·13-s + (−0.999 + 0.0335i)14-s + (−0.954 − 0.296i)16-s + (−0.239 − 0.138i)17-s + (−0.135 − 0.234i)19-s + (−0.263 + 0.103i)20-s + (0.145 − 0.0274i)22-s + (0.584 + 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.714 + 0.699i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652281 - 1.59841i\)
\(L(\frac12)\) \(\approx\) \(0.652281 - 1.59841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 + 0.921i)T \)
3 \( 1 \)
7 \( 1 + (2.06 + 1.65i)T \)
good5 \( 1 + (0.316 + 0.548i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.424 - 0.245i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.13iT - 13T^{2} \)
17 \( 1 + (0.987 + 0.569i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.591 + 1.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.05T + 29T^{2} \)
31 \( 1 + (5.40 + 3.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.53 + 3.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.06iT - 41T^{2} \)
43 \( 1 - 4.65T + 43T^{2} \)
47 \( 1 + (-4.80 - 8.32i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.10 + 1.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.12 + 5.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.1 + 5.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.11 + 5.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + (5.35 - 9.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.49 - 1.43i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (-9.12 + 5.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.17T + 97T^{2} \)
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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

−10.76403155257099769045392024970, −9.841198338176240610729465243555, −9.157154181318997884849178152155, −7.74556257840665584189958253381, −6.70756334054187641930522600727, −5.78084890233335230135800341553, −4.67004265215010783685658721828, −3.69337730979742243247762293126, −2.65384096991793220610924759596, −0.817960896890853816426906153961, 2.46482270746246093207998332479, 3.55045751359903126067730439839, 4.64273161856452469826270204965, 5.78605701589682827696477023078, 6.62040309503699052720581834288, 7.29332108896521582020714591664, 8.638536041419066671789942482228, 9.128856475184072549405080890589, 10.48933968302893037926372353709, 11.49846612494610425712242625769

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