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Properties

Label 2-504-168.101-c1-0-13
Degree $2$
Conductor $504$
Sign $0.903 - 0.428i$
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  + (−0.896 + 1.09i)2-s + (−0.391 − 1.96i)4-s + (3.16 − 1.82i)5-s + (−1.64 + 2.06i)7-s + (2.49 + 1.33i)8-s + (−0.839 + 5.09i)10-s + (−0.200 + 0.346i)11-s + 0.581·13-s + (−0.784 − 3.65i)14-s + (−3.69 + 1.53i)16-s + (2.27 − 3.94i)17-s + (2.43 + 4.21i)19-s + (−4.81 − 5.48i)20-s + (−0.199 − 0.529i)22-s + (6.21 − 3.58i)23-s + ⋯
L(s)  = 1  + (−0.634 + 0.773i)2-s + (−0.195 − 0.980i)4-s + (1.41 − 0.815i)5-s + (−0.623 + 0.782i)7-s + (0.882 + 0.470i)8-s + (−0.265 + 1.61i)10-s + (−0.0603 + 0.104i)11-s + 0.161·13-s + (−0.209 − 0.977i)14-s + (−0.923 + 0.383i)16-s + (0.552 − 0.956i)17-s + (0.557 + 0.966i)19-s + (−1.07 − 1.22i)20-s + (−0.0425 − 0.112i)22-s + (1.29 − 0.747i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.903 - 0.428i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.903 - 0.428i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23729 + 0.278714i\)
\(L(\frac12)\) \(\approx\) \(1.23729 + 0.278714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.896 - 1.09i)T \)
3 \( 1 \)
7 \( 1 + (1.64 - 2.06i)T \)
good5 \( 1 + (-3.16 + 1.82i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.200 - 0.346i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.581T + 13T^{2} \)
17 \( 1 + (-2.27 + 3.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.43 - 4.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.21 + 3.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.69T + 29T^{2} \)
31 \( 1 + (2.85 + 1.64i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.93 + 1.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.51T + 41T^{2} \)
43 \( 1 + 4.09iT - 43T^{2} \)
47 \( 1 + (-2.86 - 4.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.74 - 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.72 - 4.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.42 - 11.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.30 + 4.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.42iT - 71T^{2} \)
73 \( 1 + (6.18 + 3.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.15 - 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (5.25 + 9.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.00iT - 97T^{2} \)
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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

−10.50001030108079626104067250858, −9.810190186508856210944813674877, −9.147679222747859550812613197400, −8.595872723402140957135722947431, −7.31732323940159655143651810988, −6.22263553120623297852739561587, −5.60704694702680319098862444723, −4.82233877651021160441833303337, −2.65232500380022811495159117592, −1.20593831136745120938878564642, 1.31648872982650041981334412629, 2.74130340392112000157620410211, 3.56113095568144600040477547694, 5.18390927407177741794208634812, 6.55067920293632532361460055802, 7.10708850995970751498960714520, 8.388435642259984360744833768975, 9.517713208796955226531825118637, 9.938510927347639550501358258979, 10.70880284322632708575840179817

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