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Properties

Label 1-85-85.53-r1-0-0
Degree $1$
Conductor $85$
Sign $0.564 + 0.825i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.306900455 + 1.744140732i\)
\(L(\frac12)\) \(\approx\) \(3.306900455 + 1.744140732i\)
\(L(1)\) \(\approx\) \(2.234973436 + 0.7061374073i\)
\(L(1)\) \(\approx\) \(2.234973436 + 0.7061374073i\)

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

−30.37949726527011483859753243971, −29.56352946201280617642286049480, −28.82899249986115262470030821011, −26.80198354864851235451183767947, −25.826885276448170752436845670471, −24.83026959507889161801752512002, −23.91964861408058195547481836586, −23.02876699311268419994810879075, −21.79315992125735158432127600198, −20.59223568789344073716795839729, −19.662288220205569546105109941727, −18.835342328822274740964565326255, −16.94475797560516927502120951337, −15.9420948523297909841040665654, −14.27916791656150432749130452834, −13.94675993836388078814453022064, −12.68909151167478362807786928784, −11.72991286987807773182716123822, −10.10143157917135555912863423860, −8.4471260596983335143522506459, −6.9590735885187606061305381916, −6.249327812753090324676014694353, −4.13876931827194811027384660645, −3.10861623608053527279576182566, −1.43644379061914263667458918564, 2.32968474890305295936909797420, 3.457956661405640418945804152221, 4.748148868584009453015508750209, 6.09872388087155823967100055449, 7.65719490785394029755611514586, 9.24840086776312417597462176282, 10.38847760440593726111994776272, 11.885869098630981034143982957880, 13.03871980294941738361869021479, 14.13992717643873104463026589239, 15.40134370978321734058756182921, 15.713283978378061598142647827279, 17.316175513184322518768937038895, 19.32801183786490852736557904533, 20.01922902400411570913054640596, 21.13085135317110113969094418398, 22.19377360722073305303821461771, 22.74065357076088524610104776546, 24.43181994537158636163895765089, 25.363351680766149564240474387558, 25.98781509983757350214017751355, 27.64446895151106455123147691495, 28.527060122286135654851993800105, 30.00601408314237265171870599325, 30.803420747922941451576663099550

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