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Properties

Label 2-504-56.27-c1-0-22
Degree $2$
Conductor $504$
Sign $0.387 - 0.921i$
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 + (3.44 + 1.44i)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 + (0.921 + 0.387i)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

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.387 - 0.921i$
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: \((2,\ 504,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09769 + 1.39390i\)
\(L(\frac12)\) \(\approx\) \(2.09769 + 1.39390i\)
\(L(\frac{3}{2})\) 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

−11.26712477519203552640613105757, −10.17679286847601259024241031944, −9.201091558584540922409811775991, −8.146549604082614420630626596860, −7.42407062931123633555855860978, −6.30281692933413139658061205064, −5.44647986154397417445479579532, −4.71559368929082660341072749577, −3.33791858061871821952116070174, −1.91127313047943948071956736332, 1.57697837840932102414293755791, 2.47611480667337430201681895818, 3.94172164502611376411847981345, 5.18025650202052253659100317628, 5.73424401545523843834577393222, 6.75764102535472814123769586702, 8.309410773453508540123074701781, 9.110523310094458148893037633856, 10.17859580999847421174231159822, 10.80751145268219932604211379424

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