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Properties

Label 1-507-507.242-r0-0-0
Degree $1$
Conductor $507$
Sign $0.872 - 0.489i$
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  + (0.464 − 0.885i)2-s + (−0.568 − 0.822i)4-s + (−0.935 + 0.354i)5-s + (0.992 − 0.120i)7-s + (−0.992 + 0.120i)8-s + (−0.120 + 0.992i)10-s + (0.464 + 0.885i)11-s + (0.354 − 0.935i)14-s + (−0.354 + 0.935i)16-s + (0.120 + 0.992i)17-s + i·19-s + (0.822 + 0.568i)20-s + 22-s + 23-s + (0.748 − 0.663i)25-s + ⋯
L(s)  = 1  + (0.464 − 0.885i)2-s + (−0.568 − 0.822i)4-s + (−0.935 + 0.354i)5-s + (0.992 − 0.120i)7-s + (−0.992 + 0.120i)8-s + (−0.120 + 0.992i)10-s + (0.464 + 0.885i)11-s + (0.354 − 0.935i)14-s + (−0.354 + 0.935i)16-s + (0.120 + 0.992i)17-s + i·19-s + (0.822 + 0.568i)20-s + 22-s + 23-s + (0.748 − 0.663i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.872 - 0.489i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.872 - 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.434257281 - 0.3747983873i\)
\(L(\frac12)\) \(\approx\) \(1.434257281 - 0.3747983873i\)
\(L(1)\) \(\approx\) \(1.146284152 - 0.3833491217i\)
\(L(1)\) \(\approx\) \(1.146284152 - 0.3833491217i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.464 - 0.885i)T \)
5 \( 1 + (-0.935 + 0.354i)T \)
7 \( 1 + (0.992 - 0.120i)T \)
11 \( 1 + (0.464 + 0.885i)T \)
17 \( 1 + (0.120 + 0.992i)T \)
19 \( 1 + iT \)
23 \( 1 + T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (0.663 - 0.748i)T \)
37 \( 1 + (-0.663 + 0.748i)T \)
41 \( 1 + (-0.239 + 0.970i)T \)
43 \( 1 + (0.748 - 0.663i)T \)
47 \( 1 + (0.822 + 0.568i)T \)
53 \( 1 + (-0.120 - 0.992i)T \)
59 \( 1 + (0.935 - 0.354i)T \)
61 \( 1 + (0.120 - 0.992i)T \)
67 \( 1 + (0.822 + 0.568i)T \)
71 \( 1 + (-0.239 + 0.970i)T \)
73 \( 1 + (-0.464 - 0.885i)T \)
79 \( 1 + (0.568 - 0.822i)T \)
83 \( 1 + (0.239 + 0.970i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.935 - 0.354i)T \)
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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

−23.83405018521070807228239215750, −23.02358223916847860331538408915, −22.18065728764964038735672884847, −21.24207319826157508203301973259, −20.54415689045393694583583945169, −19.37375266941660876946390706547, −18.49005709827452138490992422865, −17.51195613371209675331872302247, −16.73550213049693315864401433265, −15.89132822409410959939079696093, −15.20719022873599219691217232419, −14.31400209581609815614578458648, −13.581160475093722430880836597221, −12.464170761954267861795034648171, −11.639523263699269255710040888767, −10.96952008953626303587551996731, −8.98542854952688534038755051, −8.72442558822211348288464701872, −7.54638718972853962104317903485, −6.97882027019214415702528655995, −5.52404953054669733638595357894, −4.851572198838118078275906873718, −3.91242831903720464789267950195, −2.85435052386480094271458291461, −0.84121616062312629616983785418, 1.22950249846597652166148898348, 2.26495971129186097392295785637, 3.65081598221125067920384621280, 4.256011055119584761605735059, 5.22530010120398582982720962646, 6.507311514674782188112363360706, 7.70910512102466124278395195416, 8.55210930081061428570882128666, 9.7978392566528468682426900733, 10.69461367083250811422482764751, 11.46128104702774171217433162513, 12.1280648072076283773823419951, 12.97540303940552213555098390988, 14.21500038658622315020793223761, 14.874328902486476451502701528190, 15.3624188766174542922213324798, 16.94155029698854396982118800762, 17.74546258383166275784874603770, 18.86877163189983078744297227114, 19.24285523434463189621581730707, 20.5397789306164460057974809158, 20.665871997108308436277504250014, 21.934200448463302521800856028287, 22.68825248629176891027967295119, 23.3870948255320351993390027187

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