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Properties

Label 2-504-168.107-c1-0-3
Degree $2$
Conductor $504$
Sign $-0.323 - 0.946i$
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  + (−1.41 + 0.0663i)2-s + (1.99 − 0.187i)4-s + (1.02 − 1.78i)5-s + (−1.24 + 2.33i)7-s + (−2.80 + 0.396i)8-s + (−1.33 + 2.58i)10-s + (−5.24 + 3.03i)11-s + 1.77i·13-s + (1.60 − 3.37i)14-s + (3.92 − 0.746i)16-s + (−0.786 + 0.453i)17-s + (−3.37 + 5.84i)19-s + (1.71 − 3.73i)20-s + (7.21 − 4.62i)22-s + (0.351 − 0.608i)23-s + ⋯
L(s)  = 1  + (−0.998 + 0.0469i)2-s + (0.995 − 0.0937i)4-s + (0.459 − 0.796i)5-s + (−0.471 + 0.882i)7-s + (−0.990 + 0.140i)8-s + (−0.421 + 0.816i)10-s + (−1.58 + 0.913i)11-s + 0.493i·13-s + (0.429 − 0.903i)14-s + (0.982 − 0.186i)16-s + (−0.190 + 0.110i)17-s + (−0.774 + 1.34i)19-s + (0.383 − 0.835i)20-s + (1.53 − 0.987i)22-s + (0.0732 − 0.126i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319109 + 0.446124i\)
\(L(\frac12)\) \(\approx\) \(0.319109 + 0.446124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0663i)T \)
3 \( 1 \)
7 \( 1 + (1.24 - 2.33i)T \)
good5 \( 1 + (-1.02 + 1.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.24 - 3.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.77iT - 13T^{2} \)
17 \( 1 + (0.786 - 0.453i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.37 - 5.84i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.351 + 0.608i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.52T + 29T^{2} \)
31 \( 1 + (-7.31 + 4.22i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.07 + 3.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 + (4.42 - 7.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.31 - 5.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.11 + 0.645i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.95 + 4.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.562 + 0.974i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.43T + 71T^{2} \)
73 \( 1 + (-2.08 - 3.61i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.6 - 6.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.71iT - 83T^{2} \)
89 \( 1 + (-4.08 - 2.35i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−10.93024958387289780265163301028, −9.981285623760076873344774842829, −9.544002487964147674668577300186, −8.468124720211134196730587855113, −7.955006995222537738463497646792, −6.62778785740554811750097048530, −5.78305191503214741419262295218, −4.75183422972611039121027037806, −2.78808589629498554589569709416, −1.77423692994409069727594684546, 0.41951043051151711443361479159, 2.51088596148929821218577374506, 3.25239827315435478596371868274, 5.15880513839487685613032963416, 6.45907276472369725740280090058, 6.96154217790816469700884555277, 8.059818340673566387606378024630, 8.792563774833805930180783808635, 10.15840889055579852157463886690, 10.44587871987873643413501380702

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