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Properties

Label 1-504-504.389-r1-0-0
Degree $1$
Conductor $504$
Sign $-0.235 - 0.971i$
Analytic cond. $54.1623$
Root an. cond. $54.1623$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s − 47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s − 47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(54.1623\)
Root analytic conductor: \(54.1623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (1:\ ),\ -0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8566292025 - 1.089190292i\)
\(L(\frac12)\) \(\approx\) \(0.8566292025 - 1.089190292i\)
\(L(1)\) \(\approx\) \(0.9422082591 - 0.2284861489i\)
\(L(1)\) \(\approx\) \(0.9422082591 - 0.2284861489i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.55205907502834663413536949110, −22.873474016014034104984707489870, −22.07001633902224739570677053729, −21.12099345531651954395176335295, −20.39933200732561779584111397764, −19.11196154208055129067156931223, −18.71140439693465151894689159448, −18.01773735223045840206225047279, −16.51969194417617592252639766686, −16.16494396461543506503216863006, −15.028157308592685874737916815671, −14.19726195469510521108595288173, −13.533964185287517945553273006200, −12.2248998468494195797801932757, −11.41545401193003913591045038373, −10.70916552989236555936316315907, −9.717934452451814432043536159145, −8.55245624957393102812957798197, −7.69111905090385165759858429263, −6.736413517094143028395458849801, −5.845589690454669806762627150804, −4.58300602695863833041283362538, −3.423849908835673469667264954331, −2.68577022306380124472164254096, −1.08139475230148576852833022749, 0.41206160183456978050020257725, 1.56444629583340275828724549163, 3.01282004947376538775826340487, 4.13992467108402251082474271010, 5.075053178104863612953463305660, 5.960535035271447104104686185284, 7.446054812638601207892589442972, 7.99369081058528418140766689273, 9.06900902366768056233797464497, 9.954249245124108774404678120375, 11.02905578119330154564938868280, 11.95831843714441469046549465115, 12.90324781351032911213657447741, 13.35376537477999411980008923882, 14.83540870422311203800260186542, 15.520966109550925637795967660274, 16.208035941521123136142843798859, 17.390570651550092721947399745434, 17.82963451839623237585425087069, 19.19448103069673339762324926848, 19.764045141134739118468898476190, 20.75979285822666264288710447452, 21.188920321696824705181282927084, 22.566963669111079627845215707020, 23.20197110323312433853873163115

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