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Properties

Label 1-507-507.266-r0-0-0
Degree $1$
Conductor $507$
Sign $-0.269 + 0.963i$
Analytic cond. $2.35449$
Root an. cond. $2.35449$
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  + (−0.774 − 0.632i)2-s + (0.200 + 0.979i)4-s + (−0.992 + 0.120i)5-s + (−0.534 − 0.845i)7-s + (0.464 − 0.885i)8-s + (0.845 + 0.534i)10-s + (−0.160 − 0.987i)11-s + (−0.120 + 0.992i)14-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (0.866 + 0.5i)19-s + (−0.316 − 0.948i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.970 − 0.239i)25-s + ⋯
L(s)  = 1  + (−0.774 − 0.632i)2-s + (0.200 + 0.979i)4-s + (−0.992 + 0.120i)5-s + (−0.534 − 0.845i)7-s + (0.464 − 0.885i)8-s + (0.845 + 0.534i)10-s + (−0.160 − 0.987i)11-s + (−0.120 + 0.992i)14-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (0.866 + 0.5i)19-s + (−0.316 − 0.948i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.970 − 0.239i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.269 + 0.963i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ -0.269 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04489256042 + 0.05916659959i\)
\(L(\frac12)\) \(\approx\) \(0.04489256042 + 0.05916659959i\)
\(L(1)\) \(\approx\) \(0.4429514130 - 0.1443718161i\)
\(L(1)\) \(\approx\) \(0.4429514130 - 0.1443718161i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.774 - 0.632i)T \)
5 \( 1 + (-0.992 + 0.120i)T \)
7 \( 1 + (-0.534 - 0.845i)T \)
11 \( 1 + (-0.160 - 0.987i)T \)
17 \( 1 + (-0.845 + 0.534i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.632 - 0.774i)T \)
31 \( 1 + (-0.239 + 0.970i)T \)
37 \( 1 + (-0.960 + 0.278i)T \)
41 \( 1 + (0.0804 - 0.996i)T \)
43 \( 1 + (-0.278 + 0.960i)T \)
47 \( 1 + (-0.663 + 0.748i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
59 \( 1 + (-0.391 + 0.919i)T \)
61 \( 1 + (-0.0402 + 0.999i)T \)
67 \( 1 + (0.979 + 0.200i)T \)
71 \( 1 + (-0.903 + 0.428i)T \)
73 \( 1 + (-0.935 - 0.354i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (-0.822 + 0.568i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.600 - 0.799i)T \)
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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

−23.47565447451976757236782634563, −22.75435554678864200755788690513, −21.9038470861998001162685673837, −20.289484579283941916546086599944, −19.9815958522950388052606315531, −19.00619663201290502704720551767, −18.260927349075957893223169330, −17.5362032599416434725126492294, −16.35483748499463972322031994492, −15.5428772185785264346933109948, −15.39125729123639805567864071796, −14.17139526844318516233269458641, −12.94199854299863079573933965186, −11.89337797679037756013147005861, −11.21006254296112301984165235842, −9.92016450076542885082860558621, −9.2196137045249438632714326833, −8.3464394346153973124558573380, −7.36075374137482837497770748114, −6.719618075217248530407467918039, −5.43458889495810567379402347053, −4.59417453477238674824190454578, −3.08473188269161364546402579207, −1.813217722332635880297225070842, −0.05740641051048744959350226148, 1.19303603140909324579119878530, 2.856990008512733633398212196397, 3.618449719407648551848448104687, 4.47340643265190596902096803033, 6.3141365061083856169710123223, 7.21894592517644132617012153388, 8.11998743584613720086566526093, 8.80222267585689401856697743307, 10.09628808710036288300968287880, 10.72569670663366707422873552595, 11.54464889207262982542134339594, 12.41735784570940277113598425444, 13.329397004970639021194368779387, 14.29941353466280652571876582061, 15.883196123981303809265502345823, 16.101866702975328648757611386432, 17.10965527802156583717877173043, 18.07538522396457872681974201394, 19.03886980721772457921160749937, 19.54650871264208457670836455914, 20.25836116535705415785689083761, 21.093037992366700742394392631370, 22.2368939976312946416397528691, 22.79445258708234214415084182036, 23.925900009980070773252079888962

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