Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $0.155 + 0.987i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + (−0.707 + 0.707i)3-s + 4-s + (−0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + 8-s i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s + i·13-s + (0.707 + 0.707i)14-s + 16-s i·18-s + i·19-s − 21-s + (−0.707 + 0.707i)22-s + ⋯
L(s,χ)  = 1  + 2-s + (−0.707 + 0.707i)3-s + 4-s + (−0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + 8-s i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s + i·13-s + (0.707 + 0.707i)14-s + 16-s i·18-s + i·19-s − 21-s + (−0.707 + 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.155 + 0.987i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.155 + 0.987i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $0.155 + 0.987i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (42, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (1:\ ),\ 0.155 + 0.987i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.937550760 + 1.656861195i$
$L(\frac12,\chi)$  $\approx$  $1.937550760 + 1.656861195i$
$L(\chi,1)$  $\approx$  1.564949276 + 0.6460131224i
$L(1,\chi)$  $\approx$  1.564949276 + 0.6460131224i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−30.2590042434739685427556716361, −29.49129888002026279673486693395, −28.54590487189018973994261292942, −27.213642052157351889310804059902, −25.6745665899582841795506737175, −24.41603578948084947941748038372, −23.85676883021435128671345300341, −22.92584614444477137184398701733, −21.902829691998247898829131192863, −20.7195089561950501425442416724, −19.627073911243945990688789043384, −18.19399062984341302588003507716, −17.06035055233939237211754307081, −15.99034490874341473581072318805, −14.56934474573974527136806468972, −13.37887786412928789884890963050, −12.69167234519937943645190867423, −11.15339311085651349735649871047, −10.7270813720592477556435840441, −8.02166544150101384489786394484, −7.04825138025576652941407084590, −5.67260459597104584014729736809, −4.70321665385727569779937376759, −2.806801890407419578575545111732, −1.01197678233887133333759179347, 2.04800666371401874744443552427, 3.92770920793898100798060838221, 5.04432515186375419126390836198, 5.968849313332163391633463896002, 7.53736108917955684147013306161, 9.464445409578840619481716280384, 10.900504376457943883277352997875, 11.74463422870077579862779685670, 12.76832911179048694804948366323, 14.42701997171482655630176186403, 15.24894566451437434672465558664, 16.236769077326309201220821061924, 17.4061803070254353691274757232, 18.80394218262535901178489583172, 20.63619479331883099619722749216, 21.173543704375009142835115748561, 22.144735900378495156193242381428, 23.195982240378959663858460662570, 23.996201309428875729031551182579, 25.24114076221577001839236839159, 26.47017491646817501029852805506, 27.8738376602431028403721014288, 28.67144621879573491312879575549, 29.62681750718925970372147813860, 31.14859964257367323526403591583

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