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Properties

Label 2-504-63.16-c1-0-7
Degree $2$
Conductor $504$
Sign $-0.219 - 0.975i$
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  + (0.577 + 1.63i)3-s + 1.85·5-s + (−2.60 + 0.464i)7-s + (−2.33 + 1.88i)9-s − 2.57·11-s + (2.82 + 4.88i)13-s + (1.07 + 3.03i)15-s + (3.57 + 6.19i)17-s + (0.636 − 1.10i)19-s + (−2.26 − 3.98i)21-s + 0.241·23-s − 1.55·25-s + (−4.42 − 2.71i)27-s + (0.923 − 1.59i)29-s + (1.49 − 2.59i)31-s + ⋯
L(s)  = 1  + (0.333 + 0.942i)3-s + 0.829·5-s + (−0.984 + 0.175i)7-s + (−0.777 + 0.628i)9-s − 0.776·11-s + (0.782 + 1.35i)13-s + (0.276 + 0.782i)15-s + (0.868 + 1.50i)17-s + (0.146 − 0.252i)19-s + (−0.493 − 0.869i)21-s + 0.0503·23-s − 0.311·25-s + (−0.852 − 0.523i)27-s + (0.171 − 0.297i)29-s + (0.268 − 0.465i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.219 - 0.975i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936468 + 1.17116i\)
\(L(\frac12)\) \(\approx\) \(0.936468 + 1.17116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.577 - 1.63i)T \)
7 \( 1 + (2.60 - 0.464i)T \)
good5 \( 1 - 1.85T + 5T^{2} \)
11 \( 1 + 2.57T + 11T^{2} \)
13 \( 1 + (-2.82 - 4.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.57 - 6.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.636 + 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.241T + 23T^{2} \)
29 \( 1 + (-0.923 + 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.49 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.338 + 0.585i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.733 + 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.14 + 7.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.15 - 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.35 - 5.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.04 - 1.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.47 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.53T + 71T^{2} \)
73 \( 1 + (6.55 + 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.86 - 3.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.00 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.60 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.40 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.80691316915002264685974342968, −10.23310603004447239373040083445, −9.410461739367851225610122664162, −8.841438151546754862814974613502, −7.70618433822879466598375541423, −6.20422988669937857558247511142, −5.74924118012723123733099885849, −4.34703135779625602065274825339, −3.36175075469709218365850908612, −2.11522313819063072654451023779, 0.874370735983960193073997141853, 2.61225407024018854885891730228, 3.34133821690469965770972880463, 5.40440581702585851023315878330, 5.98533458057103183528996571266, 7.07140069902055866974260465371, 7.84924102095041342132619191716, 8.857286210842028469214991836737, 9.833626193234039843489387166627, 10.44154440556871470027105529942

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