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Properties

Label 1-85-85.33-r1-0-0
Degree $1$
Conductor $85$
Sign $0.525 + 0.850i$
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  − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s − 11-s + i·12-s + i·13-s + 14-s + 16-s + i·18-s − 19-s + 21-s + i·22-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s − 11-s + i·12-s + i·13-s + 14-s + 16-s + i·18-s − 19-s + 21-s + i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2472304674 + 0.1378400134i\)
\(L(\frac12)\) \(\approx\) \(0.2472304674 + 0.1378400134i\)
\(L(1)\) \(\approx\) \(0.5797245932 - 0.3582895027i\)
\(L(1)\) \(\approx\) \(0.5797245932 - 0.3582895027i\)

Euler product

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

−30.61451488790641208711874013612, −29.09391322671288447540873839707, −27.70357190983460156500494632922, −27.07164003373821228707824941453, −26.05928191012093379262360852307, −25.34227811528652178963387691721, −23.66684732330415004146244980835, −23.1056612884681863849727754160, −21.90302044681689308313978630451, −20.80000427216869858184920883414, −19.564030262281343528098889078771, −17.88511901758013824567295451952, −17.03551775277582715307668626707, −15.97057813884395773064445303566, −15.159398398384385099590625654544, −13.999707406443972193259544980416, −12.89354143319032937422050106762, −10.75494154917379644807478296165, −9.97249789162520345894948852656, −8.531211828387148762050582545681, −7.43243547332462651843915068334, −5.77981487186049345904308076485, −4.68091996093687510035973857921, −3.42996445890210008910601007450, −0.128735410143107181537474885058, 1.84112232112415947823875068667, 2.83854672221219985281648384012, 4.87125573577287819897283637950, 6.31251051491627121182766042965, 8.1241661628544843181800051528, 9.029800784108006679647040039717, 10.66864669067719649183939481341, 11.8852707460841161758196610774, 12.64980275323560641834857348246, 13.69540705554533425807172333411, 14.92261744071392487991277550385, 16.80186374824049969875852559152, 18.24397105532338449462262564023, 18.65480300790086695392407665153, 19.69557398925269690603257766196, 20.981511070297628630624515460106, 21.928472713101861486506748627254, 23.22271193852470998543571786938, 24.02990196868621081330927197462, 25.41497724605685095524185700458, 26.4503410722791328831382334210, 27.97094889243822468944359617337, 28.77296194306712816905355627487, 29.44583446663783624419730077448, 30.816422420652801361915893489799

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