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Properties

Label 2-504-168.101-c1-0-1
Degree $2$
Conductor $504$
Sign $0.266 - 0.963i$
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  + (−0.169 − 1.40i)2-s + (−1.94 + 0.475i)4-s + (−1.53 + 0.887i)5-s + (0.843 − 2.50i)7-s + (0.995 + 2.64i)8-s + (1.50 + 2.00i)10-s + (−2.09 + 3.62i)11-s − 3.76·13-s + (−3.66 − 0.760i)14-s + (3.54 − 1.84i)16-s + (−2.32 + 4.02i)17-s + (0.0315 + 0.0545i)19-s + (2.56 − 2.45i)20-s + (5.44 + 2.32i)22-s + (−4.05 + 2.34i)23-s + ⋯
L(s)  = 1  + (−0.119 − 0.992i)2-s + (−0.971 + 0.237i)4-s + (−0.687 + 0.397i)5-s + (0.318 − 0.947i)7-s + (0.352 + 0.935i)8-s + (0.476 + 0.635i)10-s + (−0.631 + 1.09i)11-s − 1.04·13-s + (−0.979 − 0.203i)14-s + (0.887 − 0.461i)16-s + (−0.563 + 0.976i)17-s + (0.00723 + 0.0125i)19-s + (0.573 − 0.549i)20-s + (1.16 + 0.495i)22-s + (−0.845 + 0.488i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.266 - 0.963i$
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: \((2,\ 504,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.329174 + 0.250489i\)
\(L(\frac12)\) \(\approx\) \(0.329174 + 0.250489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.169 + 1.40i)T \)
3 \( 1 \)
7 \( 1 + (-0.843 + 2.50i)T \)
good5 \( 1 + (1.53 - 0.887i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.09 - 3.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.76T + 13T^{2} \)
17 \( 1 + (2.32 - 4.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0315 - 0.0545i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.05 - 2.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.47T + 29T^{2} \)
31 \( 1 + (-6.64 - 3.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.91 - 1.68i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.06T + 41T^{2} \)
43 \( 1 - 9.94iT - 43T^{2} \)
47 \( 1 + (0.338 + 0.586i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.36 + 4.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.53 + 3.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.23 + 12.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.67 - 3.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.10iT - 71T^{2} \)
73 \( 1 + (-1.29 - 0.746i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.99 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.64iT - 83T^{2} \)
89 \( 1 + (8.97 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.24iT - 97T^{2} \)
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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

−11.05121180600501332368240698740, −10.19647319228403920100238011376, −9.827680753849937619418434830346, −8.297757360659927250902837896977, −7.74237118901345772445338535671, −6.76100785189259957171902177948, −4.92691855535438117098481466155, −4.30696638220566709773347179962, −3.15181001479045793769174164274, −1.78598837912263658857963764715, 0.25318532133685993499727184827, 2.71306367703762849053460671869, 4.35683037582920791864205436528, 5.14224977503260599903476830056, 6.05714234249942402576504809967, 7.18166999691958839830108915654, 8.241025023240333738868021481219, 8.526435502779513169583516817548, 9.600785518460110222011874884178, 10.58731446930770975462594931468

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