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Properties

Label 2-504-63.59-c1-0-5
Degree $2$
Conductor $504$
Sign $0.914 + 0.404i$
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  + (−1.70 − 0.278i)3-s − 0.542·5-s + (−2.62 + 0.340i)7-s + (2.84 + 0.950i)9-s + 0.769i·11-s + (2.96 + 1.71i)13-s + (0.926 + 0.150i)15-s + (3.23 − 5.60i)17-s + (5.60 − 3.23i)19-s + (4.58 + 0.148i)21-s + 0.115i·23-s − 4.70·25-s + (−4.59 − 2.41i)27-s + (4.40 − 2.54i)29-s + (4.01 − 2.31i)31-s + ⋯
L(s)  = 1  + (−0.987 − 0.160i)3-s − 0.242·5-s + (−0.991 + 0.128i)7-s + (0.948 + 0.316i)9-s + 0.231i·11-s + (0.822 + 0.474i)13-s + (0.239 + 0.0389i)15-s + (0.784 − 1.35i)17-s + (1.28 − 0.742i)19-s + (0.999 + 0.0323i)21-s + 0.0241i·23-s − 0.941·25-s + (−0.885 − 0.465i)27-s + (0.817 − 0.471i)29-s + (0.720 − 0.416i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.914 + 0.404i$
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: \((2,\ 504,\ (\ :1/2),\ 0.914 + 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901121 - 0.190130i\)
\(L(\frac12)\) \(\approx\) \(0.901121 - 0.190130i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.278i)T \)
7 \( 1 + (2.62 - 0.340i)T \)
good5 \( 1 + 0.542T + 5T^{2} \)
11 \( 1 - 0.769iT - 11T^{2} \)
13 \( 1 + (-2.96 - 1.71i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.23 + 5.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.60 + 3.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.115iT - 23T^{2} \)
29 \( 1 + (-4.40 + 2.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.01 + 2.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.47 - 9.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.04 + 7.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.32 - 5.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.773 + 1.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.221 - 0.127i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.12 + 8.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.83 - 2.78i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.64 - 2.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.67iT - 71T^{2} \)
73 \( 1 + (5.35 + 3.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.01 - 3.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.80 - 10.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.00 - 3.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.0 - 8.69i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.05918533832180334260535563025, −9.792719319840093048618973313890, −9.509850491666800560563002278894, −7.957389329638838306055937162185, −7.04198074158224496102804627369, −6.26577144300264608250715693162, −5.32721636835164514495615658456, −4.21177497834897173745806888272, −2.87547564174825591998898906599, −0.853474518420224255179549787374, 1.06135931517369741372671161738, 3.31730809386634164805098786738, 4.14358276282731541581554397473, 5.74250447666861359331211095128, 6.02151096634800557546366708335, 7.25269587329758562843056407954, 8.189716128085718227904273496650, 9.476206431801392715790600296082, 10.22929237533848317370810450953, 10.86225022700486240063396137837

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