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Properties

Label 2-504-168.101-c1-0-18
Degree $2$
Conductor $504$
Sign $0.881 + 0.472i$
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  + (−1.40 − 0.127i)2-s + (1.96 + 0.359i)4-s + (1.87 − 1.08i)5-s + (2.55 + 0.669i)7-s + (−2.72 − 0.758i)8-s + (−2.77 + 1.28i)10-s + (1.67 − 2.89i)11-s + 1.74·13-s + (−3.51 − 1.27i)14-s + (3.74 + 1.41i)16-s + (−0.283 + 0.491i)17-s + (−0.270 − 0.467i)19-s + (4.07 − 1.45i)20-s + (−2.72 + 3.86i)22-s + (−5.21 + 3.01i)23-s + ⋯
L(s)  = 1  + (−0.995 − 0.0903i)2-s + (0.983 + 0.179i)4-s + (0.837 − 0.483i)5-s + (0.967 + 0.253i)7-s + (−0.963 − 0.268i)8-s + (−0.877 + 0.405i)10-s + (0.503 − 0.872i)11-s + 0.483·13-s + (−0.940 − 0.339i)14-s + (0.935 + 0.354i)16-s + (−0.0688 + 0.119i)17-s + (−0.0619 − 0.107i)19-s + (0.910 − 0.324i)20-s + (−0.580 + 0.823i)22-s + (−1.08 + 0.627i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.881 + 0.472i$
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: \((2,\ 504,\ (\ :1/2),\ 0.881 + 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18217 - 0.297126i\)
\(L(\frac12)\) \(\approx\) \(1.18217 - 0.297126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.127i)T \)
3 \( 1 \)
7 \( 1 + (-2.55 - 0.669i)T \)
good5 \( 1 + (-1.87 + 1.08i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.67 + 2.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.74T + 13T^{2} \)
17 \( 1 + (0.283 - 0.491i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.270 + 0.467i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.21 - 3.01i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.77T + 29T^{2} \)
31 \( 1 + (6.56 + 3.79i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.60 + 5.54i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.77T + 41T^{2} \)
43 \( 1 - 1.80iT - 43T^{2} \)
47 \( 1 + (-0.679 - 1.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.46 + 2.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.84 - 5.68i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.60 + 9.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.7 + 6.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.79iT - 71T^{2} \)
73 \( 1 + (-7.56 - 4.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.77 + 8.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.9iT - 83T^{2} \)
89 \( 1 + (6.30 + 10.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.4iT - 97T^{2} \)
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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

−10.97749760017023383327709798976, −9.713404718850945562050821655329, −9.120996071343508100867969064502, −8.322050786808555257658407551004, −7.53237643263802216040970657624, −6.10600761486083659750138581644, −5.60958808487587870396104085719, −3.94056628900772601888058826694, −2.29237409246094589423192807395, −1.21477577274253163169828381047, 1.48743169964490963109449858606, 2.48294234592600286457171470297, 4.25437052265798851608917101221, 5.69771840658342174857394443914, 6.55109668541690386311842245128, 7.45622736118693583522110370446, 8.337254880331970242355134294595, 9.285757820769499726954715179971, 10.11018214959266827209341129152, 10.72864386639742602945674830570

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