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Properties

Label 2-504-63.47-c1-0-21
Degree $2$
Conductor $504$
Sign $0.340 + 0.940i$
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  + (0.958 − 1.44i)3-s + 4.11·5-s + (−2.54 − 0.711i)7-s + (−1.16 − 2.76i)9-s − 5.82i·11-s + (−2.52 + 1.45i)13-s + (3.94 − 5.93i)15-s + (1.58 + 2.73i)17-s + (0.722 + 0.417i)19-s + (−3.47 + 2.99i)21-s + 7.09i·23-s + 11.9·25-s + (−5.10 − 0.978i)27-s + (−1.91 − 1.10i)29-s + (3.66 + 2.11i)31-s + ⋯
L(s)  = 1  + (0.553 − 0.832i)3-s + 1.84·5-s + (−0.963 − 0.269i)7-s + (−0.386 − 0.922i)9-s − 1.75i·11-s + (−0.699 + 0.403i)13-s + (1.01 − 1.53i)15-s + (0.383 + 0.664i)17-s + (0.165 + 0.0956i)19-s + (−0.757 + 0.653i)21-s + 1.47i·23-s + 2.38·25-s + (−0.982 − 0.188i)27-s + (−0.355 − 0.205i)29-s + (0.658 + 0.380i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64263 - 1.15248i\)
\(L(\frac12)\) \(\approx\) \(1.64263 - 1.15248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.958 + 1.44i)T \)
7 \( 1 + (2.54 + 0.711i)T \)
good5 \( 1 - 4.11T + 5T^{2} \)
11 \( 1 + 5.82iT - 11T^{2} \)
13 \( 1 + (2.52 - 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.58 - 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.722 - 0.417i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.09iT - 23T^{2} \)
29 \( 1 + (1.91 + 1.10i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.66 - 2.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.82 + 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.04 - 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.155 + 0.269i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.502 - 0.870i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.94 - 1.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.51 + 4.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.98 - 2.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.99 + 8.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (3.04 - 1.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.579 - 1.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.57 - 13.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.82 + 8.35i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.06 - 2.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.56417452657668962445488325126, −9.588191524709051073157925026974, −9.201670404993165283071654569898, −8.099828020138655930954505061514, −6.90488705809814316465382287436, −6.09370986583053058107407985745, −5.59548015828246294223885951001, −3.49158865083488025858259457137, −2.57293269892992171035011169773, −1.24673588221104025868202550275, 2.19926590157922097111840892762, 2.84245284867371565936781750594, 4.57066670622107001596198934618, 5.34389421115866834813458345002, 6.40087895568089434002467849380, 7.39430028519589929135801137754, 8.856463856630243630952332208701, 9.606416223683676408058567056841, 9.946436075285452622953795293405, 10.51358184111449001215872652047

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