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Properties

Label 2-504-63.59-c1-0-2
Degree $2$
Conductor $504$
Sign $0.0246 - 0.999i$
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  + (−0.498 − 1.65i)3-s − 0.623·5-s + (−0.996 + 2.45i)7-s + (−2.50 + 1.65i)9-s + 5.19i·11-s + (−2.74 − 1.58i)13-s + (0.310 + 1.03i)15-s + (0.437 − 0.757i)17-s + (−1.41 + 0.819i)19-s + (4.56 + 0.431i)21-s + 8.25i·23-s − 4.61·25-s + (3.99 + 3.32i)27-s + (4.96 − 2.86i)29-s + (4.02 − 2.32i)31-s + ⋯
L(s)  = 1  + (−0.287 − 0.957i)3-s − 0.278·5-s + (−0.376 + 0.926i)7-s + (−0.834 + 0.551i)9-s + 1.56i·11-s + (−0.760 − 0.438i)13-s + (0.0802 + 0.267i)15-s + (0.106 − 0.183i)17-s + (−0.325 + 0.187i)19-s + (0.995 + 0.0941i)21-s + 1.72i·23-s − 0.922·25-s + (0.768 + 0.640i)27-s + (0.921 − 0.532i)29-s + (0.722 − 0.417i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0246 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0246 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0246 - 0.999i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0246 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.466496 + 0.455152i\)
\(L(\frac12)\) \(\approx\) \(0.466496 + 0.455152i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.498 + 1.65i)T \)
7 \( 1 + (0.996 - 2.45i)T \)
good5 \( 1 + 0.623T + 5T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + (2.74 + 1.58i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.437 + 0.757i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 - 0.819i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.25iT - 23T^{2} \)
29 \( 1 + (-4.96 + 2.86i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.02 + 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.24 - 2.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.52 - 6.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.56 + 2.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.73 - 8.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.15 + 0.665i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.18 + 5.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.65 - 5.57i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.04 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (11.6 + 6.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.84 - 8.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.192 + 0.332i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.0198 - 0.0344i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.94 + 3.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.63052936412683964555176208258, −10.07719115100118720955957045021, −9.497870531876126113459240405623, −8.166522834299726461988600950840, −7.54858613999314046302000060574, −6.60643447487838538294993702584, −5.64789601985045867124419132122, −4.65626030294239531033514799428, −2.92074457712396090606945857179, −1.83271672408578468478596158204, 0.38873517735882656761185811723, 2.95432581827649686126499041830, 3.94935957405491755760442072212, 4.81391132733342634487542809375, 6.05545866044129847543357385575, 6.87025569515380686140015216770, 8.244198143607794755151486631896, 8.905585285357290490673636433377, 10.12577121579484815708567139682, 10.51719938987790833714963013161

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