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Properties

Label 2-504-56.27-c1-0-37
Degree $2$
Conductor $504$
Sign $-0.921 - 0.387i$
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 − i)2-s − 2i·4-s − 2.44·5-s + (−2.44 + i)7-s + (−2 − 2i)8-s + (−2.44 + 2.44i)10-s − 2·11-s − 2.44·13-s + (−1.44 + 3.44i)14-s − 4·16-s − 4.89i·17-s + 2.44i·19-s + 4.89i·20-s + (−2 + 2i)22-s − 4i·23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s − 1.09·5-s + (−0.925 + 0.377i)7-s + (−0.707 − 0.707i)8-s + (−0.774 + 0.774i)10-s − 0.603·11-s − 0.679·13-s + (−0.387 + 0.921i)14-s − 16-s − 1.18i·17-s + 0.561i·19-s + 1.09i·20-s + (−0.426 + 0.426i)22-s − 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.921 - 0.387i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105713 + 0.524462i\)
\(L(\frac12)\) \(\approx\) \(0.105713 + 0.524462i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
7 \( 1 + (2.44 - i)T \)
good5 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 2.44iT - 59T^{2} \)
61 \( 1 + 7.34T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−10.52527879228554239123502578135, −9.769578558879480856245606856214, −8.819390362562797854401500560947, −7.57160465744598826660731986486, −6.64869439425792830369763702995, −5.46982820528701773359454494584, −4.50957778156980327147369271345, −3.40639756910428476918577160733, −2.51913206292206919270441881376, −0.23782929495710966055307941386, 2.85771347036401477959112211020, 3.82404186127485442425258286208, 4.69107014025028315820919613273, 5.93532279355091335703683474430, 6.88040285896038654558016114487, 7.69395612298119901418674434224, 8.346434473743140385391052267003, 9.580386189880353483519488950257, 10.63002838866197089874893195260, 11.73612418976197146953590075844

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