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Properties

Label 2-504-56.3-c1-0-28
Degree $2$
Conductor $504$
Sign $0.704 + 0.709i$
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  + (1.30 − 0.554i)2-s + (1.38 − 1.44i)4-s + (1.03 + 1.80i)5-s + (−1.25 − 2.33i)7-s + (1.00 − 2.64i)8-s + (2.35 + 1.76i)10-s + (0.669 − 1.16i)11-s + 2.50·13-s + (−2.92 − 2.33i)14-s + (−0.167 − 3.99i)16-s + (2.78 + 1.60i)17-s + (3.55 − 2.05i)19-s + (4.03 + 0.991i)20-s + (0.227 − 1.88i)22-s + (−5.54 + 3.20i)23-s + ⋯
L(s)  = 1  + (0.919 − 0.392i)2-s + (0.692 − 0.721i)4-s + (0.464 + 0.805i)5-s + (−0.473 − 0.880i)7-s + (0.353 − 0.935i)8-s + (0.743 + 0.558i)10-s + (0.201 − 0.349i)11-s + 0.694·13-s + (−0.780 − 0.624i)14-s + (−0.0417 − 0.999i)16-s + (0.674 + 0.389i)17-s + (0.815 − 0.470i)19-s + (0.902 + 0.221i)20-s + (0.0485 − 0.401i)22-s + (−1.15 + 0.668i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40808 - 1.00236i\)
\(L(\frac12)\) \(\approx\) \(2.40808 - 1.00236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.554i)T \)
3 \( 1 \)
7 \( 1 + (1.25 + 2.33i)T \)
good5 \( 1 + (-1.03 - 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.669 + 1.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.50T + 13T^{2} \)
17 \( 1 + (-2.78 - 1.60i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.55 + 2.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.54 - 3.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.66iT - 29T^{2} \)
31 \( 1 + (2.21 - 3.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.50 - 3.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.55iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (0.565 + 0.980i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.43 + 4.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.29 + 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 + 4.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.93 + 6.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (-0.480 - 0.277i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.26 - 3.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.503iT - 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−10.79436588217323735531967635840, −10.23754652304915864230511888735, −9.421064951624725100676407324952, −7.84888979259181366497913958343, −6.77735218279799923669492906098, −6.22071749735035760622499629953, −5.12022432213111895189229257505, −3.69859665879483744456582680550, −3.12936160229383291021284991047, −1.46144729850965766509815834561, 1.90156239332271749289396222339, 3.27059751813144827570898410036, 4.44662179971872161159202207924, 5.63827983076739455277959074644, 5.96429837479427322204472008606, 7.26980654601261606448983124029, 8.294475812411858335746359826319, 9.154003081610621113779682928097, 10.05793691320511853416419781796, 11.37635590360474976106940867034

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