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Properties

Label 2-504-8.5-c1-0-10
Degree $2$
Conductor $504$
Sign $-i$
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  + 1.41i·2-s − 2.00·4-s − 1.41i·5-s + 7-s − 2.82i·8-s + 2.00·10-s + 2.82i·11-s + 4.24i·13-s + 1.41i·14-s + 4.00·16-s + 6·17-s + 4.24i·19-s + 2.82i·20-s − 4.00·22-s + 6·23-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 0.632i·5-s + 0.377·7-s − 1.00i·8-s + 0.632·10-s + 0.852i·11-s + 1.17i·13-s + 0.377i·14-s + 1.00·16-s + 1.45·17-s + 0.973i·19-s + 0.632i·20-s − 0.852·22-s + 1.25·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951411 + 0.951411i\)
\(L(\frac12)\) \(\approx\) \(0.951411 + 0.951411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 8.48iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−11.11437400160768083366740731815, −9.877216842551701482527123592479, −9.253161678510047364510449930347, −8.360725636778285381215973336333, −7.48552588621373163826546283391, −6.68407790506238318675937481741, −5.42271579476445323548618189862, −4.77505236191553178206315729560, −3.65943890560118029816277891516, −1.44664328594197053453024460775, 0.978967310572132702839073127709, 2.80761367022671508585790385587, 3.41706763986660724589316201429, 4.94486771904310955657934264019, 5.73897235843596944810450150580, 7.21235993344681281482262024124, 8.213700238104133155937845540303, 9.002321149579703297555318913908, 10.16349232710368384616591770146, 10.69663240111516261178934131882

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