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Properties

Label 2-504-24.11-c1-0-20
Degree $2$
Conductor $504$
Sign $0.929 + 0.369i$
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  + (1.40 − 0.111i)2-s + (1.97 − 0.315i)4-s + 0.472·5-s − i·7-s + (2.74 − 0.665i)8-s + (0.665 − 0.0528i)10-s − 3.86i·11-s + 3.00i·13-s + (−0.111 − 1.40i)14-s + (3.80 − 1.24i)16-s + 1.23i·17-s + 6.30·19-s + (0.932 − 0.148i)20-s + (−0.431 − 5.44i)22-s − 1.60·23-s + ⋯
L(s)  = 1  + (0.996 − 0.0790i)2-s + (0.987 − 0.157i)4-s + 0.211·5-s − 0.377i·7-s + (0.971 − 0.235i)8-s + (0.210 − 0.0167i)10-s − 1.16i·11-s + 0.832i·13-s + (−0.0298 − 0.376i)14-s + (0.950 − 0.311i)16-s + 0.298i·17-s + 1.44·19-s + (0.208 − 0.0333i)20-s + (−0.0920 − 1.16i)22-s − 0.333·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.929 + 0.369i$
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: \((2,\ 504,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71436 - 0.519248i\)
\(L(\frac12)\) \(\approx\) \(2.71436 - 0.519248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.111i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 0.472T + 5T^{2} \)
11 \( 1 + 3.86iT - 11T^{2} \)
13 \( 1 - 3.00iT - 13T^{2} \)
17 \( 1 - 1.23iT - 17T^{2} \)
19 \( 1 - 6.30T + 19T^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 + 6.36T + 29T^{2} \)
31 \( 1 - 9.09iT - 31T^{2} \)
37 \( 1 + 0.844iT - 37T^{2} \)
41 \( 1 + 2.07iT - 41T^{2} \)
43 \( 1 - 5.97T + 43T^{2} \)
47 \( 1 + 7.22T + 47T^{2} \)
53 \( 1 + 6.64T + 53T^{2} \)
59 \( 1 + 7.22iT - 59T^{2} \)
61 \( 1 - 7.67iT - 61T^{2} \)
67 \( 1 - 0.304T + 67T^{2} \)
71 \( 1 + 4.28T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 8.86iT - 79T^{2} \)
83 \( 1 + 9.28iT - 83T^{2} \)
89 \( 1 - 16.1iT - 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−11.14848219446079893183973871580, −10.18874379459996126860844767037, −9.196498029457529390780841839677, −7.940855864997146869812188005407, −7.03253170758593327823664967268, −6.04198058191698341399397317003, −5.25690204976663577552423800661, −4.00468359422471015397306786269, −3.13023545165098779083706764013, −1.56560749223865590805089078404, 1.90801943491758002835827759983, 3.09832359514246311207739736596, 4.32364995577583154327451594223, 5.37434094476935743694454973977, 6.05792107528079014092513647135, 7.37015991304250556125225276267, 7.84262067111361487954758994872, 9.460597118573995956276380804099, 10.07059611347265682963963353932, 11.30039074490627816183300623643

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