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Properties

Label 1-517-517.14-r0-0-0
Degree $1$
Conductor $517$
Sign $-0.411 - 0.911i$
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  + (−0.662 + 0.749i)2-s + (0.641 + 0.767i)3-s + (−0.122 − 0.992i)4-s + (−0.230 + 0.973i)5-s + (−0.999 − 0.0273i)6-s + (−0.740 + 0.672i)7-s + (0.824 + 0.565i)8-s + (−0.176 + 0.984i)9-s + (−0.576 − 0.816i)10-s + (0.682 − 0.730i)12-s + (0.0409 − 0.999i)13-s + (−0.0136 − 0.999i)14-s + (−0.894 + 0.447i)15-s + (−0.969 + 0.243i)16-s + (−0.839 − 0.542i)17-s + (−0.620 − 0.784i)18-s + ⋯
L(s)  = 1  + (−0.662 + 0.749i)2-s + (0.641 + 0.767i)3-s + (−0.122 − 0.992i)4-s + (−0.230 + 0.973i)5-s + (−0.999 − 0.0273i)6-s + (−0.740 + 0.672i)7-s + (0.824 + 0.565i)8-s + (−0.176 + 0.984i)9-s + (−0.576 − 0.816i)10-s + (0.682 − 0.730i)12-s + (0.0409 − 0.999i)13-s + (−0.0136 − 0.999i)14-s + (−0.894 + 0.447i)15-s + (−0.969 + 0.243i)16-s + (−0.839 − 0.542i)17-s + (−0.620 − 0.784i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.411 - 0.911i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ -0.411 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2530664903 + 0.3917287722i\)
\(L(\frac12)\) \(\approx\) \(-0.2530664903 + 0.3917287722i\)
\(L(1)\) \(\approx\) \(0.4222056919 + 0.5106893482i\)
\(L(1)\) \(\approx\) \(0.4222056919 + 0.5106893482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.662 + 0.749i)T \)
3 \( 1 + (0.641 + 0.767i)T \)
5 \( 1 + (-0.230 + 0.973i)T \)
7 \( 1 + (-0.740 + 0.672i)T \)
13 \( 1 + (0.0409 - 0.999i)T \)
17 \( 1 + (-0.839 - 0.542i)T \)
19 \( 1 + (-0.385 + 0.922i)T \)
23 \( 1 + (-0.917 + 0.398i)T \)
29 \( 1 + (-0.620 - 0.784i)T \)
31 \( 1 + (0.927 + 0.373i)T \)
37 \( 1 + (-0.0136 + 0.999i)T \)
41 \( 1 + (-0.282 - 0.959i)T \)
43 \( 1 + (0.682 + 0.730i)T \)
53 \( 1 + (0.792 + 0.609i)T \)
59 \( 1 + (-0.981 + 0.190i)T \)
61 \( 1 + (-0.955 - 0.295i)T \)
67 \( 1 + (0.203 + 0.979i)T \)
71 \( 1 + (-0.662 - 0.749i)T \)
73 \( 1 + (0.721 + 0.692i)T \)
79 \( 1 + (-0.702 - 0.711i)T \)
83 \( 1 + (0.360 - 0.932i)T \)
89 \( 1 + (0.854 + 0.519i)T \)
97 \( 1 + (-0.969 - 0.243i)T \)
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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

−23.07725512391645405149753688908, −21.87353249151540363085006904541, −20.98583849925626488515192976844, −20.08085962315887445310012149151, −19.720758935173827809511584063545, −19.05852981346438829603963279874, −18.031276560870535084965126375128, −17.15163999078059774211964797234, −16.45964483979049247724146159810, −15.479877746956811244299596754467, −13.94383253807420789309805067127, −13.2674847809975239322132766896, −12.6403413079462724729249184071, −11.847077660225476942541583671951, −10.80153599447522844290100131581, −9.53189930637175992912527684287, −8.9915831782573028812691260807, −8.18508742391805914409312077094, −7.211486344844267959963055762993, −6.38141135774377612042774811827, −4.40246379036714421514648560336, −3.76564307695554558360907160818, −2.44729748023295705648366870782, −1.50080967815940346106434138672, −0.277122490771231537739357194090, 2.17007603834742746968132235409, 3.08670405570092420709169813689, 4.26137272206684137303154803987, 5.61046750023417764905910119654, 6.35307712034407850613033143512, 7.55391120073877053333995193709, 8.268790179932529883692991272315, 9.25779392507408381162365053229, 10.07600783903599650613716044061, 10.61743422725595049805019803242, 11.796185209143625291046276518150, 13.36051096993227462761123663408, 14.14173833545324268906030164878, 15.16859107368374244684442525609, 15.47234672342692118676990809485, 16.18760686534042215133021175374, 17.333026184043261870146065929964, 18.31637886799307598922614805802, 19.011723705754624581918555058068, 19.70813186778617666481558863914, 20.53798124157581692909815351521, 21.83241530352482796282551346854, 22.57939986356033962850727364539, 23.044274509430963455394307004795, 24.554287867780421689587497541647

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