Properties

Degree 1
Conductor 311
Sign $-0.999 + 0.00822i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.612 + 0.790i)2-s + (−0.612 − 0.790i)3-s + (−0.250 − 0.968i)4-s + (0.528 + 0.848i)5-s + 6-s + (−0.0506 + 0.998i)7-s + (0.918 + 0.394i)8-s + (−0.250 + 0.968i)9-s + (−0.994 − 0.101i)10-s + (0.440 + 0.897i)11-s + (−0.612 + 0.790i)12-s + (−0.250 + 0.968i)13-s + (−0.758 − 0.651i)14-s + (0.347 − 0.937i)15-s + (−0.874 + 0.485i)16-s + (0.994 + 0.101i)17-s + ⋯
L(s,χ)  = 1  + (−0.612 + 0.790i)2-s + (−0.612 − 0.790i)3-s + (−0.250 − 0.968i)4-s + (0.528 + 0.848i)5-s + 6-s + (−0.0506 + 0.998i)7-s + (0.918 + 0.394i)8-s + (−0.250 + 0.968i)9-s + (−0.994 − 0.101i)10-s + (0.440 + 0.897i)11-s + (−0.612 + 0.790i)12-s + (−0.250 + 0.968i)13-s + (−0.758 − 0.651i)14-s + (0.347 − 0.937i)15-s + (−0.874 + 0.485i)16-s + (0.994 + 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.00822i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.00822i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $-0.999 + 0.00822i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (142, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ -0.999 + 0.00822i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.003634317088 + 0.8831955817i$
$L(\frac12,\chi)$  $\approx$  $0.003634317088 + 0.8831955817i$
$L(\chi,1)$  $\approx$  0.5687170541 + 0.3910101680i
$L(1,\chi)$  $\approx$  0.5687170541 + 0.3910101680i

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

−24.65479302466909111975471115417, −23.47425263134412117404275368855, −22.57574988392109743026968629930, −21.5008058004347366791618535968, −21.1446359439783114579496189737, −20.054127945016688831862970984437, −19.55056271671556109028511752370, −17.85266823473967609026341098120, −17.44551998421166861305964181298, −16.51808016192866943016543919506, −16.0641027086473869015305092975, −14.323871486870354820827608162698, −13.29233620744892142476811224379, −12.36756782310460724735061493294, −11.39198140421395175530827553554, −10.41451820173033303122691361383, −9.85133531147015688790792366716, −8.86159045634113672343308051664, −7.85338379666865164539214170190, −6.30358879015115271628411992799, −5.057041004024938986654683672181, −4.084901875283629765924605962332, −3.04173374698834638099969397608, −1.13005653505012804153433055814, −0.40871565650559372805289275966, 1.54178063382814826413468298526, 2.36320718295061384506866032892, 4.6212846145441077565526419094, 5.96695768726666699578062047631, 6.31951913826507778265116847399, 7.342308559438131689915846184965, 8.31659217512971739214476236883, 9.63138370571609136030433349456, 10.32417104095919620886656544403, 11.65251785845045935319147562161, 12.40833427733499095558150180451, 13.91029184236643244391748734192, 14.47579601080691462331926638209, 15.49445108207081099565407298782, 16.68698034051163521356955722729, 17.39801957785710047143194229812, 18.23908452305984827202445659926, 18.85637382023450275294321022453, 19.44399579176828885564888992507, 21.148683454432283664669708586391, 22.314718910329947832645334801837, 22.85447898568465054726552984768, 23.80812102891659049637591103614, 24.79393674890534778684374515205, 25.39501376012982555995268257179

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