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Properties

Label 1-85-85.59-r0-0-0
Degree $1$
Conductor $85$
Sign $-0.758 + 0.651i$
Analytic cond. $0.394738$
Root an. cond. $0.394738$
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 + (0.707 + 0.707i)3-s − 4-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)7-s − i·8-s + i·9-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + 13-s + (−0.707 − 0.707i)14-s + 16-s − 18-s − i·19-s − 21-s + (−0.707 − 0.707i)22-s + ⋯
L(s)  = 1  + i·2-s + (0.707 + 0.707i)3-s − 4-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)7-s − i·8-s + i·9-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + 13-s + (−0.707 − 0.707i)14-s + 16-s − 18-s − i·19-s − 21-s + (−0.707 − 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.758 + 0.651i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ -0.758 + 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3478100082 + 0.9390469205i\)
\(L(\frac12)\) \(\approx\) \(0.3478100082 + 0.9390469205i\)
\(L(1)\) \(\approx\) \(0.7247234035 + 0.7805983208i\)
\(L(1)\) \(\approx\) \(0.7247234035 + 0.7805983208i\)

Euler product

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

−30.23500195191550261560026617795, −29.39509310169605385573837598812, −28.68682765339910678957518402838, −27.062381473753784812984000438482, −26.289596487660277066939515421519, −25.21479007283342120040149316056, −23.56439350028804470796122679442, −23.07959510594324804485289633813, −21.380101606176422853421870699839, −20.555301046161030601588965866672, −19.49989043946104381519144907248, −18.768593283216669130822520053480, −17.73310425563286755887464957169, −16.15390594983896790226582134186, −14.38470149494576533213962054849, −13.39734352081556036573853212695, −12.80294391030712362857679594749, −11.29966036407139295891687605980, −10.06883415304507147641414773101, −8.81970526940337293071379741961, −7.697982717932381577176910422939, −5.98000514122303056276591833240, −3.85636013568059355256340769581, −2.897568113447931841807196569285, −1.15832363681212858763789260095, 2.85299293783222917403203356982, 4.36653930811977800866970174663, 5.63105648180611671401780699636, 7.13669291021869687880779248388, 8.55818630834904285081944796824, 9.31055153271395306910335205710, 10.56552429175653442477372506210, 12.74156991647335659659540656092, 13.674006835809029951478155382576, 15.07424001522578746235243674691, 15.62738890902798858471001769511, 16.59034032318196149491474047020, 18.13047037985524952869834069843, 19.09160126422064896851795081173, 20.49265375517073911169627846266, 21.73367146430478683024153394307, 22.61664495636251718846292089532, 23.80343196942068078837831826185, 25.27012312206083330041252889268, 25.69517564849393624004259681484, 26.605781005458273664641505627924, 27.840848339610381907698202117756, 28.599034147741302117630571229023, 30.76070546012272868482751214682, 31.30456446563388292500143978901

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