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Properties

Label 1-85-85.42-r1-0-0
Degree $1$
Conductor $85$
Sign $0.155 + 0.987i$
Analytic cond. $9.13451$
Root an. cond. $9.13451$
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  + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ 0.155 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.937550760 + 1.656861195i\)
\(L(\frac12)\) \(\approx\) \(1.937550760 + 1.656861195i\)
\(L(1)\) \(\approx\) \(1.564949276 + 0.6460131224i\)
\(L(1)\) \(\approx\) \(1.564949276 + 0.6460131224i\)

Euler product

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

−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