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Properties

Label 2-504-24.11-c1-0-21
Degree $2$
Conductor $504$
Sign $0.344 + 0.938i$
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  + (0.937 − 1.05i)2-s + (−0.242 − 1.98i)4-s + 4.34·5-s + i·7-s + (−2.32 − 1.60i)8-s + (4.06 − 4.59i)10-s − 1.12i·11-s − 0.491i·13-s + (1.05 + 0.937i)14-s + (−3.88 + 0.961i)16-s + 5.96i·17-s + 2.34·19-s + (−1.05 − 8.61i)20-s + (−1.19 − 1.05i)22-s − 6.68·23-s + ⋯
L(s)  = 1  + (0.662 − 0.748i)2-s + (−0.121 − 0.992i)4-s + 1.94·5-s + 0.377i·7-s + (−0.823 − 0.567i)8-s + (1.28 − 1.45i)10-s − 0.339i·11-s − 0.136i·13-s + (0.282 + 0.250i)14-s + (−0.970 + 0.240i)16-s + 1.44i·17-s + 0.537·19-s + (−0.235 − 1.92i)20-s + (−0.254 − 0.225i)22-s − 1.39·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09410 - 1.46177i\)
\(L(\frac12)\) \(\approx\) \(2.09410 - 1.46177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.937 + 1.05i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 4.34T + 5T^{2} \)
11 \( 1 + 1.12iT - 11T^{2} \)
13 \( 1 + 0.491iT - 13T^{2} \)
17 \( 1 - 5.96iT - 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
23 \( 1 + 6.68T + 23T^{2} \)
29 \( 1 + 4.70T + 29T^{2} \)
31 \( 1 + 8.07iT - 31T^{2} \)
37 \( 1 - 0.750iT - 37T^{2} \)
41 \( 1 + 3.41iT - 41T^{2} \)
43 \( 1 + 7.30T + 43T^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 - 9.47iT - 59T^{2} \)
61 \( 1 - 1.64iT - 61T^{2} \)
67 \( 1 - 9.48T + 67T^{2} \)
71 \( 1 - 9.14T + 71T^{2} \)
73 \( 1 - 2.65T + 73T^{2} \)
79 \( 1 - 5.74iT - 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 + 7.51T + 97T^{2} \)
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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

−10.65406786601220229381415721751, −9.937100551237873195592606846407, −9.412011916603628807042309485250, −8.327385853377410688756252127786, −6.51863888327421772383663880401, −5.85788831943697915169223374453, −5.28778252458336269973836586801, −3.81096232254371836505139929949, −2.43874664860723037279620190651, −1.61778119598145828697188028331, 1.97026809417910447274918173820, 3.22709052901088354623763898895, 4.83203024161499489383916713741, 5.44033572707712430391749109723, 6.46357656082080398493726218611, 7.09054518624445058005722664175, 8.324659823884872714159684207914, 9.496713849669538124217596359221, 9.847140189449920423653953629289, 11.14552330051123636237229248026

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