Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $-0.758 + 0.651i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $-0.758 + 0.651i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (59, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (0:\ ),\ -0.758 + 0.651i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3478100082 + 0.9390469205i$
$L(\frac12,\chi)$  $\approx$  $0.3478100082 + 0.9390469205i$
$L(\chi,1)$  $\approx$  0.7247234035 + 0.7805983208i
$L(1,\chi)$  $\approx$  0.7247234035 + 0.7805983208i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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