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Properties

Label 1-85-85.57-r0-0-0
Degree $1$
Conductor $85$
Sign $-0.724 - 0.688i$
Analytic cond. $0.394738$
Root an. cond. $0.394738$
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  + (−0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s − i·4-s + (0.923 + 0.382i)6-s + (−0.923 − 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.923 + 0.382i)12-s − 13-s + (0.923 − 0.382i)14-s − 16-s − i·18-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 − 0.382i)22-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s − i·4-s + (0.923 + 0.382i)6-s + (−0.923 − 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.923 + 0.382i)12-s − 13-s + (0.923 − 0.382i)14-s − 16-s − i·18-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 − 0.382i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.724 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.724 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.724 - 0.688i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ -0.724 - 0.688i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08928106553 - 0.2235191158i\)
\(L(\frac12)\) \(\approx\) \(0.08928106553 - 0.2235191158i\)
\(L(1)\) \(\approx\) \(0.4294688579 - 0.1020763067i\)
\(L(1)\) \(\approx\) \(0.4294688579 - 0.1020763067i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 - T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (-0.382 - 0.923i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.382 - 0.923i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−31.40134044194414453549314904721, −29.55826032559495271943083024882, −28.984276359289998788994577742464, −28.01238570633121481938327570702, −27.11310135232674559400304153313, −26.10807539844722781299942126105, −25.333580252382846115729520199, −23.344105792424552006714056172924, −22.23516912783899901982705972547, −21.485476647705395588432827675047, −20.40612520185525379009281912081, −19.359042920622371440340294804285, −18.15871832681635481511565606828, −16.97898274557538364724628110947, −16.11950886094157234681238326527, −14.97990612886508400396814252090, −12.97225653484040312459631900597, −12.04824537621092211911797925209, −10.68214302782606843679258552689, −9.85596342223925887134491441122, −8.9231071825530929938929691276, −7.27328334579718767942258940877, −5.45789470828319650454728115623, −3.85705240619494951152021901707, −2.54528249262928664135182877806, 0.31808636057728767729118627245, 2.43525560267084227011274805788, 5.09060132704338165328197960169, 6.40218057799490864212948522330, 7.26698260075404946332182931229, 8.43943994609804439044077628646, 9.94953776400973228622195903742, 11.06874636715782241608524832818, 12.73027214438804170281741796777, 13.68599868644918342182133685245, 15.12394455564625498691665335873, 16.489040350333895067217739805935, 17.16794331678531674442694160403, 18.43100124549418401493144166779, 19.1867255434862138607227569429, 20.13479190047071827161064413313, 22.17126986148278219061676782728, 23.3002627955444567759545672829, 24.01033708281403746228697970690, 25.081445234739382006271453557, 26.03222196707254168069712146046, 26.98961235354758927103751802464, 28.49737564867443819906327725773, 29.028106680099927868205628653130, 30.02943377449248199554825787503

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