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Properties

Label 1-311-311.198-r1-0-0
Degree $1$
Conductor $311$
Sign $-0.482 + 0.875i$
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  + (−0.874 + 0.485i)2-s + (−0.874 − 0.485i)3-s + (0.528 − 0.848i)4-s + (−0.612 − 0.790i)5-s + 6-s + (0.979 + 0.201i)7-s + (−0.0506 + 0.998i)8-s + (0.528 + 0.848i)9-s + (0.918 + 0.394i)10-s + (0.250 + 0.968i)11-s + (−0.874 + 0.485i)12-s + (0.528 + 0.848i)13-s + (−0.954 + 0.299i)14-s + (0.151 + 0.988i)15-s + (−0.440 − 0.897i)16-s + (−0.918 − 0.394i)17-s + ⋯
L(s)  = 1  + (−0.874 + 0.485i)2-s + (−0.874 − 0.485i)3-s + (0.528 − 0.848i)4-s + (−0.612 − 0.790i)5-s + 6-s + (0.979 + 0.201i)7-s + (−0.0506 + 0.998i)8-s + (0.528 + 0.848i)9-s + (0.918 + 0.394i)10-s + (0.250 + 0.968i)11-s + (−0.874 + 0.485i)12-s + (0.528 + 0.848i)13-s + (−0.954 + 0.299i)14-s + (0.151 + 0.988i)15-s + (−0.440 − 0.897i)16-s + (−0.918 − 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.482 + 0.875i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ -0.482 + 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2046723985 + 0.3466325780i\)
\(L(\frac12)\) \(\approx\) \(0.2046723985 + 0.3466325780i\)
\(L(1)\) \(\approx\) \(0.5116657533 + 0.04149772258i\)
\(L(1)\) \(\approx\) \(0.5116657533 + 0.04149772258i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.874 + 0.485i)T \)
3 \( 1 + (-0.874 - 0.485i)T \)
5 \( 1 + (-0.612 - 0.790i)T \)
7 \( 1 + (0.979 + 0.201i)T \)
11 \( 1 + (0.250 + 0.968i)T \)
13 \( 1 + (0.528 + 0.848i)T \)
17 \( 1 + (-0.918 - 0.394i)T \)
19 \( 1 + (0.612 - 0.790i)T \)
23 \( 1 + (0.0506 - 0.998i)T \)
29 \( 1 + (0.954 + 0.299i)T \)
31 \( 1 + (-0.918 + 0.394i)T \)
37 \( 1 + (-0.979 + 0.201i)T \)
41 \( 1 + (-0.347 + 0.937i)T \)
43 \( 1 + (-0.979 - 0.201i)T \)
47 \( 1 + (-0.954 - 0.299i)T \)
53 \( 1 + (0.979 + 0.201i)T \)
59 \( 1 + (-0.979 - 0.201i)T \)
61 \( 1 + (0.612 - 0.790i)T \)
67 \( 1 + (0.347 + 0.937i)T \)
71 \( 1 + (-0.820 + 0.571i)T \)
73 \( 1 + (-0.994 - 0.101i)T \)
79 \( 1 + (0.347 - 0.937i)T \)
83 \( 1 + (0.151 - 0.988i)T \)
89 \( 1 + (0.979 - 0.201i)T \)
97 \( 1 + (-0.820 + 0.571i)T \)
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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

−24.706727655539342952169567901185, −23.80412978916052634119469412182, −22.7507635904667156503856230859, −21.9194623321877123227437328693, −21.16930555755772413337569911338, −20.21890762596374248188667486488, −19.22065499275366743159042383933, −18.13722337714387618145969513426, −17.776264150422366918194290405943, −16.69884308996306535094976108178, −15.78102061114064505598858712374, −15.05535706189885614401969235345, −13.62022588528107601449543330497, −12.14786309969876219251498872857, −11.37618108496414089048773413631, −10.86846303582083467140284061509, −10.13208795478577129494919754981, −8.71186252496408732480943872986, −7.83387482124146718279186248597, −6.7659590169998918244020836654, −5.6142496912890002354166007107, −4.02766286874136339503931088002, −3.2793335881471338032970407756, −1.48414436069342611082357910697, −0.21318905946978977600880289230, 1.12734881054134220730716777057, 1.98472603461892463782991529564, 4.63686089407130861783071449980, 5.05986811459664399215584469297, 6.59130278451441297062017511436, 7.25553354197895079273546580539, 8.3653038204334585551750058595, 9.10905388618314528073199230615, 10.51288274658760456621280155002, 11.54571261295658316444417094536, 11.92694419294261534897197818885, 13.31171990373980851248097272959, 14.58242825595768420982509488500, 15.69118556295865407180856701331, 16.34158327325673020422535848005, 17.307710874728367389359748757569, 17.94828767013553380245424169059, 18.69216818699989791368807212646, 19.86643644550365661023370545188, 20.49059662791545141060436587734, 21.77287755467470141022246408985, 23.140632141417388413097173118541, 23.677422032085342289032400380995, 24.5555206851939102331902650273, 24.91728457361716271460659454843

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