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Properties

Label 1-517-517.102-r0-0-0
Degree $1$
Conductor $517$
Sign $0.299 - 0.954i$
Analytic cond. $2.40093$
Root an. cond. $2.40093$
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  + (0.905 − 0.423i)2-s + (0.721 − 0.692i)3-s + (0.641 − 0.767i)4-s + (−0.702 + 0.711i)5-s + (0.360 − 0.932i)6-s + (0.507 + 0.861i)7-s + (0.256 − 0.966i)8-s + (0.0409 − 0.999i)9-s + (−0.334 + 0.942i)10-s + (−0.0682 − 0.997i)12-s + (−0.230 − 0.973i)13-s + (0.824 + 0.565i)14-s + (−0.0136 + 0.999i)15-s + (−0.176 − 0.984i)16-s + (0.994 − 0.109i)17-s + (−0.385 − 0.922i)18-s + ⋯
L(s)  = 1  + (0.905 − 0.423i)2-s + (0.721 − 0.692i)3-s + (0.641 − 0.767i)4-s + (−0.702 + 0.711i)5-s + (0.360 − 0.932i)6-s + (0.507 + 0.861i)7-s + (0.256 − 0.966i)8-s + (0.0409 − 0.999i)9-s + (−0.334 + 0.942i)10-s + (−0.0682 − 0.997i)12-s + (−0.230 − 0.973i)13-s + (0.824 + 0.565i)14-s + (−0.0136 + 0.999i)15-s + (−0.176 − 0.984i)16-s + (0.994 − 0.109i)17-s + (−0.385 − 0.922i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.299 - 0.954i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ 0.299 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.308504515 - 1.694845262i\)
\(L(\frac12)\) \(\approx\) \(2.308504515 - 1.694845262i\)
\(L(1)\) \(\approx\) \(1.902300015 - 0.8604283477i\)
\(L(1)\) \(\approx\) \(1.902300015 - 0.8604283477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.905 - 0.423i)T \)
3 \( 1 + (0.721 - 0.692i)T \)
5 \( 1 + (-0.702 + 0.711i)T \)
7 \( 1 + (0.507 + 0.861i)T \)
13 \( 1 + (-0.230 - 0.973i)T \)
17 \( 1 + (0.994 - 0.109i)T \)
19 \( 1 + (0.149 + 0.988i)T \)
23 \( 1 + (0.682 - 0.730i)T \)
29 \( 1 + (-0.385 - 0.922i)T \)
31 \( 1 + (-0.435 + 0.900i)T \)
37 \( 1 + (0.824 - 0.565i)T \)
41 \( 1 + (0.998 + 0.0546i)T \)
43 \( 1 + (-0.0682 + 0.997i)T \)
53 \( 1 + (-0.839 + 0.542i)T \)
59 \( 1 + (-0.531 + 0.847i)T \)
61 \( 1 + (0.792 - 0.609i)T \)
67 \( 1 + (-0.917 + 0.398i)T \)
71 \( 1 + (0.905 + 0.423i)T \)
73 \( 1 + (-0.620 - 0.784i)T \)
79 \( 1 + (-0.955 + 0.295i)T \)
83 \( 1 + (-0.868 + 0.496i)T \)
89 \( 1 + (0.460 + 0.887i)T \)
97 \( 1 + (-0.176 + 0.984i)T \)
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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

−23.8767468481668881554738082739, −23.02706511624246564074933214539, −21.91114696773289368533964733106, −21.203250678628717988558753817, −20.49225717501773401030217445675, −19.91693382217799504533502911822, −19.008840109712687996792272799310, −17.25150668992532641728802899754, −16.65188318954157478252709121729, −15.99384912216566887707728649353, −15.046274364716476293151094936615, −14.41253018011614979510950032690, −13.56299962654292749417115841339, −12.79826092912648153880579809379, −11.546031353901442862168668706086, −11.00925007457194964470017209554, −9.5695011420746606945572070279, −8.62522641714838271761737960180, −7.65715167477482752242266533216, −7.14821001273469450132584571778, −5.40629868580860784103108396247, −4.62499160459157721408124995147, −3.988708280778877280230971997638, −3.10273508125721370404954556302, −1.62145829954054245270973376036, 1.218412169187075052844634397434, 2.543378254089617113188944616169, 3.06473713943351527952013209318, 4.11874662163322548021197256461, 5.47988814419104053270765546907, 6.33043718196621583181624702086, 7.534820018314028726847734740977, 8.0570844197118041165814670843, 9.448553243888007391666444545834, 10.548848897604698536185366358648, 11.52801119541915574718933789816, 12.331865770002563205703089598143, 12.825474176738115683719614734138, 14.18262206643230030294688919546, 14.66846441181188865892434061494, 15.199463123282682988579305388433, 16.20581537864208918659490221762, 17.88054781796224819872856166290, 18.68279785007685974485643568066, 19.15466494961353711453724417894, 20.09046466889451557199615789160, 20.84626175407380022193937300311, 21.63293406915323736178125710140, 22.82547297530351415115775678393, 23.13689296477238878969141249799

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