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Properties

Label 1-85-85.38-r1-0-0
Degree $1$
Conductor $85$
Sign $-0.109 + 0.994i$
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 + 3-s − 4-s + i·6-s + 7-s − i·8-s + 9-s + i·11-s − 12-s + i·13-s + i·14-s + 16-s + i·18-s + 19-s + 21-s − 22-s + ⋯
L(s)  = 1  + i·2-s + 3-s − 4-s + i·6-s + 7-s − i·8-s + 9-s + i·11-s − 12-s + i·13-s + i·14-s + 16-s + i·18-s + 19-s + 21-s − 22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.109 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.109 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.109 + 0.994i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ -0.109 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552661560 + 1.732427409i\)
\(L(\frac12)\) \(\approx\) \(1.552661560 + 1.732427409i\)
\(L(1)\) \(\approx\) \(1.280511764 + 0.8261655281i\)
\(L(1)\) \(\approx\) \(1.280511764 + 0.8261655281i\)

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 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
19 \( 1 + T \)
23 \( 1 - iT \)
29 \( 1 + T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 - T \)
43 \( 1 + iT \)
47 \( 1 + iT \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 \)
67 \( 1 + iT \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - iT \)
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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

−30.29486194121357449613716011506, −29.47005473174143308108260159039, −27.976830804091730242585522989792, −27.0919152680202705147148463556, −26.359988200715468281610971869738, −24.82871291814530971702850907667, −23.91505715474301855040720936170, −22.35081382329642929394100652564, −21.288063807860990291137562085528, −20.52493551116370326675774938024, −19.59981074255785622594569064337, −18.50444644712295458374048674126, −17.582837755424516229660120260083, −15.706536224782504511878204855703, −14.29365613541970192209324307922, −13.69700981263997455540675680727, −12.34928711571515755526733759065, −11.05944395348454049385952772907, −9.92103509076293537279364771249, −8.584946354600817400219905585867, −7.83622832974468421772966160594, −5.35548632544261363862956244815, −3.86956913939863321187231323650, −2.67418230872795891753279086185, −1.176106853049164131082101619958, 1.7796478929075105273809192022, 3.92784700689315787262061837594, 5.0301142574548057662150230069, 6.93863189475755872031893417487, 7.842274551943836783887965299907, 8.948330872556021501096954856943, 10.010817843026440351879006374973, 12.05987586363985860154755497514, 13.554501827623410741556858237116, 14.38998405906814853255589924472, 15.19149240079811775147974278314, 16.37080730771282145380492382137, 17.79268643485250911168033825395, 18.5832965845912108990197495173, 19.99962026473324583163539164302, 21.117205427773709251693657394644, 22.2753198339994617903299746002, 23.80553442031721015080153133895, 24.4121846718429285823387925658, 25.524378011040799976324987832, 26.32691247833549452842053799059, 27.26580858313880318351309456040, 28.28877940412033120418798060713, 30.219028757455193982071587274057, 31.10165256550767751335865771973

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