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Properties

Label 2-504-24.11-c1-0-2
Degree $2$
Conductor $504$
Sign $-0.918 - 0.394i$
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  + (−0.619 + 1.27i)2-s + (−1.23 − 1.57i)4-s − 0.753·5-s + i·7-s + (2.76 − 0.590i)8-s + (0.467 − 0.958i)10-s + 4.40i·11-s − 4.28i·13-s + (−1.27 − 0.619i)14-s + (−0.963 + 3.88i)16-s + 7.48i·17-s + 0.157·19-s + (0.928 + 1.18i)20-s + (−5.59 − 2.72i)22-s − 2.84·23-s + ⋯
L(s)  = 1  + (−0.438 + 0.898i)2-s + (−0.616 − 0.787i)4-s − 0.337·5-s + 0.377i·7-s + (0.977 − 0.208i)8-s + (0.147 − 0.302i)10-s + 1.32i·11-s − 1.18i·13-s + (−0.339 − 0.165i)14-s + (−0.240 + 0.970i)16-s + 1.81i·17-s + 0.0360·19-s + (0.207 + 0.265i)20-s + (−1.19 − 0.581i)22-s − 0.592·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.918 - 0.394i$
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: \((2,\ 504,\ (\ :1/2),\ -0.918 - 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139844 + 0.680574i\)
\(L(\frac12)\) \(\approx\) \(0.139844 + 0.680574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.619 - 1.27i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 0.753T + 5T^{2} \)
11 \( 1 - 4.40iT - 11T^{2} \)
13 \( 1 + 4.28iT - 13T^{2} \)
17 \( 1 - 7.48iT - 17T^{2} \)
19 \( 1 - 0.157T + 19T^{2} \)
23 \( 1 + 2.84T + 23T^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 - 9.13iT - 31T^{2} \)
37 \( 1 - 6.38iT - 37T^{2} \)
41 \( 1 - 2.93iT - 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 - 6.17T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 6.17iT - 59T^{2} \)
61 \( 1 + 9.34iT - 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 - 2.69T + 71T^{2} \)
73 \( 1 - 1.74T + 73T^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 - 8.29iT - 83T^{2} \)
89 \( 1 + 5.13iT - 89T^{2} \)
97 \( 1 - 5.83T + 97T^{2} \)
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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

−11.02604078781168144450338198625, −10.18471809820480667182390715141, −9.532491616088246555345429481433, −8.286756741759320165645894970259, −7.902030420275174867928383982676, −6.79493334513059985293748936941, −5.85804420009155458877033339905, −4.90583847394755695733378879356, −3.73968211418310691983399841450, −1.76411412004303460547606762557, 0.48331166068812345472946623691, 2.25157468598940211757275104033, 3.56543931050632047677628514349, 4.38825268212074545855561432512, 5.77375608602258764343375992090, 7.22156448191883709070314478798, 7.903086997404616246073480871749, 9.076985537783183921054287891609, 9.513799111351910200011357608863, 10.69707985823576297541467706655

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