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Properties

Label 2-504-56.53-c1-0-31
Degree $2$
Conductor $504$
Sign $0.909 + 0.416i$
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 − 0.192i)2-s + (1.92 − 0.539i)4-s + (2.93 − 1.69i)5-s + (−1.85 + 1.88i)7-s + (2.59 − 1.12i)8-s + (3.78 − 2.93i)10-s + (−0.0932 − 0.0538i)11-s + 1.50i·13-s + (−2.23 + 3.00i)14-s + (3.41 − 2.07i)16-s + (0.214 − 0.372i)17-s + (−4.32 + 2.49i)19-s + (4.73 − 4.84i)20-s + (−0.141 − 0.0575i)22-s + (−4.56 − 7.90i)23-s + ⋯
L(s)  = 1  + (0.990 − 0.136i)2-s + (0.962 − 0.269i)4-s + (1.31 − 0.757i)5-s + (−0.700 + 0.713i)7-s + (0.917 − 0.398i)8-s + (1.19 − 0.929i)10-s + (−0.0281 − 0.0162i)11-s + 0.416i·13-s + (−0.596 + 0.802i)14-s + (0.854 − 0.519i)16-s + (0.0520 − 0.0902i)17-s + (−0.992 + 0.573i)19-s + (1.05 − 1.08i)20-s + (−0.0300 − 0.0122i)22-s + (−0.951 − 1.64i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.909 + 0.416i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.909 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.89171 - 0.630872i\)
\(L(\frac12)\) \(\approx\) \(2.89171 - 0.630872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.192i)T \)
3 \( 1 \)
7 \( 1 + (1.85 - 1.88i)T \)
good5 \( 1 + (-2.93 + 1.69i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.0932 + 0.0538i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.50iT - 13T^{2} \)
17 \( 1 + (-0.214 + 0.372i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.32 - 2.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.56 + 7.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.95iT - 29T^{2} \)
31 \( 1 + (0.393 - 0.680i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.68 + 4.43i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 - 6.65iT - 43T^{2} \)
47 \( 1 + (-2.87 - 4.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.286 + 0.165i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.63 + 4.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.76 - 1.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.79 - 1.61i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.72T + 71T^{2} \)
73 \( 1 + (4.38 - 7.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.785 + 1.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.33iT - 83T^{2} \)
89 \( 1 + (3.62 + 6.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 19.0T + 97T^{2} \)
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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

−10.85372999944598019308953680063, −10.03269224643095587063149967914, −9.244102416191843696135146947713, −8.275570897393105853207572665264, −6.63185549733201566297482922883, −6.11252656692182750842357137970, −5.26441801448189604270488288317, −4.24857955635965061912771833291, −2.73710343449704910596371960273, −1.75818088696094413764600185619, 1.99994134900456801178006882501, 3.08693605530009431404416705190, 4.17478496095510953919622340580, 5.57829555456455810382371909938, 6.24246723436946789965681380630, 6.96640061732843872074742409088, 7.973744157866257814709126959093, 9.584376547962596180550439003180, 10.20108404385854301396776287526, 10.92414740571419897923967762747

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