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Properties

Label 1-311-311.142-r1-0-0
Degree $1$
Conductor $311$
Sign $-0.999 + 0.00822i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
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  + (−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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.999 + 0.00822i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ -0.999 + 0.00822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003634317088 + 0.8831955817i\)
\(L(\frac12)\) \(\approx\) \(0.003634317088 + 0.8831955817i\)
\(L(1)\) \(\approx\) \(0.5687170541 + 0.3910101680i\)
\(L(1)\) \(\approx\) \(0.5687170541 + 0.3910101680i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.612 + 0.790i)T \)
3 \( 1 + (-0.612 - 0.790i)T \)
5 \( 1 + (0.528 + 0.848i)T \)
7 \( 1 + (-0.0506 + 0.998i)T \)
11 \( 1 + (0.440 + 0.897i)T \)
13 \( 1 + (-0.250 + 0.968i)T \)
17 \( 1 + (0.994 + 0.101i)T \)
19 \( 1 + (-0.528 + 0.848i)T \)
23 \( 1 + (-0.918 - 0.394i)T \)
29 \( 1 + (0.758 - 0.651i)T \)
31 \( 1 + (0.994 - 0.101i)T \)
37 \( 1 + (0.0506 + 0.998i)T \)
41 \( 1 + (0.954 - 0.299i)T \)
43 \( 1 + (0.0506 - 0.998i)T \)
47 \( 1 + (-0.758 + 0.651i)T \)
53 \( 1 + (-0.0506 + 0.998i)T \)
59 \( 1 + (0.0506 - 0.998i)T \)
61 \( 1 + (-0.528 + 0.848i)T \)
67 \( 1 + (-0.954 - 0.299i)T \)
71 \( 1 + (-0.151 - 0.988i)T \)
73 \( 1 + (0.688 + 0.724i)T \)
79 \( 1 + (-0.954 + 0.299i)T \)
83 \( 1 + (0.347 + 0.937i)T \)
89 \( 1 + (-0.0506 - 0.998i)T \)
97 \( 1 + (-0.151 - 0.988i)T \)
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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

−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