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Properties

Label 1-311-311.287-r1-0-0
Degree $1$
Conductor $311$
Sign $0.994 - 0.106i$
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  + (−0.440 + 0.897i)2-s + (−0.440 − 0.897i)3-s + (−0.612 − 0.790i)4-s + (−0.874 + 0.485i)5-s + 6-s + (0.688 − 0.724i)7-s + (0.979 − 0.201i)8-s + (−0.612 + 0.790i)9-s + (−0.0506 − 0.998i)10-s + (−0.528 − 0.848i)11-s + (−0.440 + 0.897i)12-s + (−0.612 + 0.790i)13-s + (0.347 + 0.937i)14-s + (0.820 + 0.571i)15-s + (−0.250 + 0.968i)16-s + (0.0506 + 0.998i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.897i)2-s + (−0.440 − 0.897i)3-s + (−0.612 − 0.790i)4-s + (−0.874 + 0.485i)5-s + 6-s + (0.688 − 0.724i)7-s + (0.979 − 0.201i)8-s + (−0.612 + 0.790i)9-s + (−0.0506 − 0.998i)10-s + (−0.528 − 0.848i)11-s + (−0.440 + 0.897i)12-s + (−0.612 + 0.790i)13-s + (0.347 + 0.937i)14-s + (0.820 + 0.571i)15-s + (−0.250 + 0.968i)16-s + (0.0506 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.994 - 0.106i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.994 - 0.106i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7308147364 - 0.03891752665i\)
\(L(\frac12)\) \(\approx\) \(0.7308147364 - 0.03891752665i\)
\(L(1)\) \(\approx\) \(0.5902126315 + 0.06368108180i\)
\(L(1)\) \(\approx\) \(0.5902126315 + 0.06368108180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.440 + 0.897i)T \)
3 \( 1 + (-0.440 - 0.897i)T \)
5 \( 1 + (-0.874 + 0.485i)T \)
7 \( 1 + (0.688 - 0.724i)T \)
11 \( 1 + (-0.528 - 0.848i)T \)
13 \( 1 + (-0.612 + 0.790i)T \)
17 \( 1 + (0.0506 + 0.998i)T \)
19 \( 1 + (0.874 + 0.485i)T \)
23 \( 1 + (-0.979 + 0.201i)T \)
29 \( 1 + (-0.347 + 0.937i)T \)
31 \( 1 + (0.0506 - 0.998i)T \)
37 \( 1 + (-0.688 - 0.724i)T \)
41 \( 1 + (-0.151 + 0.988i)T \)
43 \( 1 + (-0.688 + 0.724i)T \)
47 \( 1 + (0.347 - 0.937i)T \)
53 \( 1 + (0.688 - 0.724i)T \)
59 \( 1 + (-0.688 + 0.724i)T \)
61 \( 1 + (0.874 + 0.485i)T \)
67 \( 1 + (0.151 + 0.988i)T \)
71 \( 1 + (0.758 - 0.651i)T \)
73 \( 1 + (0.918 - 0.394i)T \)
79 \( 1 + (0.151 - 0.988i)T \)
83 \( 1 + (0.820 - 0.571i)T \)
89 \( 1 + (0.688 + 0.724i)T \)
97 \( 1 + (0.758 - 0.651i)T \)
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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

−25.17003398505996272594913915272, −24.00513843177778355823825400722, −22.82528944634479043025976619616, −22.332276415163404357617255392041, −21.284659485855052945804903502812, −20.35898710130758483176661306756, −20.13266575531394518702783264804, −18.66310545930825368768213563344, −17.82482240903964460559397542372, −17.09951512698055974219048802415, −15.8161738511343361966249610069, −15.38646730873814049992884281344, −14.01803392866564960837228772806, −12.410540313993481435882122051046, −12.013768582262459607061145656006, −11.17615580771252576522502542118, −10.127705682029408130045951105190, −9.29858694559922056740240546198, −8.30062985065670948437662124102, −7.41220183663509550182582769996, −5.14638568624529474855407732977, −4.84738338107840441347080054754, −3.553006033630465814396641815158, −2.39604474046346327430835642760, −0.62227257675994439771739213394, 0.49600631184650079371800990538, 1.80172030910750913038688938245, 3.77819212158951197180915338236, 5.0477984125123565501700142627, 6.13541442196785879722469873638, 7.17757500661588272872401974714, 7.78716999094458708321469300561, 8.47759934456685613597576803213, 10.1922437706562279330772126165, 11.05868802900404862314124122850, 11.90688696438405974268821100735, 13.309321234546085486330473133677, 14.16337790324300342991584135896, 14.86725642641358352911251207145, 16.272079942152063796137520898478, 16.724879416492682421767189639185, 17.87893310711794044944522413428, 18.50089299086923513733331973527, 19.33794281736671392688500749688, 20.03333569246943362480428165846, 21.755886465296528680328021167698, 22.73547521743736111880750789706, 23.69328124273184505719249121358, 23.96486274197531207547912978138, 24.67075900604883198654033126199

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