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Properties

Label 1-311-311.185-r1-0-0
Degree $1$
Conductor $311$
Sign $-0.571 - 0.820i$
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  + (0.151 − 0.988i)2-s + (0.151 + 0.988i)3-s + (−0.954 − 0.299i)4-s + (0.347 − 0.937i)5-s + 6-s + (−0.250 + 0.968i)7-s + (−0.440 + 0.897i)8-s + (−0.954 + 0.299i)9-s + (−0.874 − 0.485i)10-s + (0.758 − 0.651i)11-s + (0.151 − 0.988i)12-s + (−0.954 + 0.299i)13-s + (0.918 + 0.394i)14-s + (0.979 + 0.201i)15-s + (0.820 + 0.571i)16-s + (0.874 + 0.485i)17-s + ⋯
L(s)  = 1  + (0.151 − 0.988i)2-s + (0.151 + 0.988i)3-s + (−0.954 − 0.299i)4-s + (0.347 − 0.937i)5-s + 6-s + (−0.250 + 0.968i)7-s + (−0.440 + 0.897i)8-s + (−0.954 + 0.299i)9-s + (−0.874 − 0.485i)10-s + (0.758 − 0.651i)11-s + (0.151 − 0.988i)12-s + (−0.954 + 0.299i)13-s + (0.918 + 0.394i)14-s + (0.979 + 0.201i)15-s + (0.820 + 0.571i)16-s + (0.874 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.571 - 0.820i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ -0.571 - 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6350252480 - 1.215275606i\)
\(L(\frac12)\) \(\approx\) \(0.6350252480 - 1.215275606i\)
\(L(1)\) \(\approx\) \(0.9446375881 - 0.4037695781i\)
\(L(1)\) \(\approx\) \(0.9446375881 - 0.4037695781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.151 - 0.988i)T \)
3 \( 1 + (0.151 + 0.988i)T \)
5 \( 1 + (0.347 - 0.937i)T \)
7 \( 1 + (-0.250 + 0.968i)T \)
11 \( 1 + (0.758 - 0.651i)T \)
13 \( 1 + (-0.954 + 0.299i)T \)
17 \( 1 + (0.874 + 0.485i)T \)
19 \( 1 + (-0.347 - 0.937i)T \)
23 \( 1 + (0.440 - 0.897i)T \)
29 \( 1 + (-0.918 + 0.394i)T \)
31 \( 1 + (0.874 - 0.485i)T \)
37 \( 1 + (0.250 + 0.968i)T \)
41 \( 1 + (0.0506 - 0.998i)T \)
43 \( 1 + (0.250 - 0.968i)T \)
47 \( 1 + (0.918 - 0.394i)T \)
53 \( 1 + (-0.250 + 0.968i)T \)
59 \( 1 + (0.250 - 0.968i)T \)
61 \( 1 + (-0.347 - 0.937i)T \)
67 \( 1 + (-0.0506 - 0.998i)T \)
71 \( 1 + (-0.688 - 0.724i)T \)
73 \( 1 + (-0.612 - 0.790i)T \)
79 \( 1 + (-0.0506 + 0.998i)T \)
83 \( 1 + (0.979 - 0.201i)T \)
89 \( 1 + (-0.250 - 0.968i)T \)
97 \( 1 + (-0.688 - 0.724i)T \)
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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

−25.18951819517932386763284346015, −24.70430534431889975557462768238, −23.34219125696081418899848019780, −23.014818261590459697712653351028, −22.25204400386952134730744571209, −20.90915176284446549319063602992, −19.54583725518563028914029074440, −18.9716337505401210881636149170, −17.79456146131185050847569709759, −17.35400637459172852077126425362, −16.46733222448577771459707912736, −14.81511721081476623998347064042, −14.503058233988130643083818769877, −13.61334992921203191742948673468, −12.73965860262893637309667913078, −11.70023069453312769866012992486, −10.13411718096685137218757330283, −9.38008955395565536066355286355, −7.72537268106445845450786757944, −7.351228379794577251531597576389, −6.51520110708220259484050898001, −5.56611962916022165207167787749, −3.99759007071166991772874265927, −2.87668209433686295206588135297, −1.206605303911243154664003864163, 0.41184563506904581442230624808, 2.059092634098319288358862949292, 3.11768005224015532692933249356, 4.32022085889708259262024722059, 5.15295279585885045473365153440, 6.02302421230632881360570832244, 8.36700740477240064554123041883, 9.077507019138382911845256835649, 9.60128389510756227584320926543, 10.704532698534986188545864721139, 11.84215483056918596339760786697, 12.44544876648774988718487387256, 13.63262468499858377364143642554, 14.56921160904232951171439455642, 15.43747493912137880564329544280, 16.83946243162217799218389768901, 17.16855325795850169433750136491, 18.79690344909833622364404761039, 19.46748083021636489047298227968, 20.41718452926123105047615752562, 21.16841269316600311428500853032, 21.976772761901657208318934753452, 22.268990145822548031847666086125, 23.75468094897443428183622400143, 24.678344934063992266606763965653

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