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Properties

Label 1-311-311.260-r0-0-0
Degree $1$
Conductor $311$
Sign $-0.671 - 0.740i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (−0.250 − 0.968i)2-s + (−0.250 + 0.968i)3-s + (−0.874 + 0.485i)4-s + (−0.440 + 0.897i)5-s + 6-s + (−0.994 − 0.101i)7-s + (0.688 + 0.724i)8-s + (−0.874 − 0.485i)9-s + (0.979 + 0.201i)10-s + (−0.612 + 0.790i)11-s + (−0.250 − 0.968i)12-s + (−0.874 − 0.485i)13-s + (0.151 + 0.988i)14-s + (−0.758 − 0.651i)15-s + (0.528 − 0.848i)16-s + (0.979 + 0.201i)17-s + ⋯
L(s)  = 1  + (−0.250 − 0.968i)2-s + (−0.250 + 0.968i)3-s + (−0.874 + 0.485i)4-s + (−0.440 + 0.897i)5-s + 6-s + (−0.994 − 0.101i)7-s + (0.688 + 0.724i)8-s + (−0.874 − 0.485i)9-s + (0.979 + 0.201i)10-s + (−0.612 + 0.790i)11-s + (−0.250 − 0.968i)12-s + (−0.874 − 0.485i)13-s + (0.151 + 0.988i)14-s + (−0.758 − 0.651i)15-s + (0.528 − 0.848i)16-s + (0.979 + 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.671 - 0.740i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (260, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ -0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05859619273 - 0.1322118690i\)
\(L(\frac12)\) \(\approx\) \(0.05859619273 - 0.1322118690i\)
\(L(1)\) \(\approx\) \(0.4948738611 + 0.01709635361i\)
\(L(1)\) \(\approx\) \(0.4948738611 + 0.01709635361i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.250 - 0.968i)T \)
3 \( 1 + (-0.250 + 0.968i)T \)
5 \( 1 + (-0.440 + 0.897i)T \)
7 \( 1 + (-0.994 - 0.101i)T \)
11 \( 1 + (-0.612 + 0.790i)T \)
13 \( 1 + (-0.874 - 0.485i)T \)
17 \( 1 + (0.979 + 0.201i)T \)
19 \( 1 + (-0.440 - 0.897i)T \)
23 \( 1 + (0.688 + 0.724i)T \)
29 \( 1 + (0.151 - 0.988i)T \)
31 \( 1 + (0.979 - 0.201i)T \)
37 \( 1 + (-0.994 + 0.101i)T \)
41 \( 1 + (0.820 - 0.571i)T \)
43 \( 1 + (-0.994 - 0.101i)T \)
47 \( 1 + (0.151 - 0.988i)T \)
53 \( 1 + (-0.994 - 0.101i)T \)
59 \( 1 + (-0.994 - 0.101i)T \)
61 \( 1 + (-0.440 - 0.897i)T \)
67 \( 1 + (0.820 + 0.571i)T \)
71 \( 1 + (-0.954 + 0.299i)T \)
73 \( 1 + (-0.0506 + 0.998i)T \)
79 \( 1 + (0.820 - 0.571i)T \)
83 \( 1 + (-0.758 + 0.651i)T \)
89 \( 1 + (-0.994 + 0.101i)T \)
97 \( 1 + (-0.954 + 0.299i)T \)
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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

−25.26272484265050851815698895809, −24.714269397748894458801476858234, −23.8690039590923205250776198497, −23.23841185254602793979434726414, −22.47940532980723001162322489694, −21.146283635670742647551667321337, −19.677410112162970688638203198349, −19.085078505794741164858520320880, −18.484311786616322388614725900012, −17.102431583967106356468269180486, −16.5867630228974290824161499958, −15.93951006612802908721381853556, −14.57873536969545654235744936962, −13.63955670223734258760489895080, −12.69279610124783638188459725519, −12.18171635958429461040702173168, −10.55882112255323947067907598878, −9.33257076504445328578834700669, −8.39067049548901971312344447732, −7.62938167376506349435273031141, −6.6046126784988955554993882892, −5.69066082849005450210644513831, −4.758905967766100731591100290727, −3.12165834766601194622304188048, −1.20423712526021180076282084056, 0.11709314660068523526235831419, 2.60042207863496545903832773777, 3.238710474724018364538507545470, 4.30082970086716112305868913705, 5.40454631463691952556842081525, 6.95414815651478489129229392574, 8.08596197834175317838593591729, 9.53936777637910996886189358391, 10.05143254314664247948604266678, 10.746734230654079808424726781778, 11.80936029701271756701235006602, 12.62680006057314318492970513410, 13.83838495711342045234986198991, 15.05840459428603684875901364582, 15.6465664021472407139873540923, 17.02721219460599960827731703043, 17.61656520330826867468713082985, 18.92512435094554075483037649548, 19.51906379434059546100703738831, 20.41352922914083001085062654903, 21.42638511888344517707611145198, 22.12141599896786269327033569558, 23.0352591033243093488853285870, 23.227086364983489078095565640828, 25.43139361839598384305419117591

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