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Properties

Label 1-311-311.143-r1-0-0
Degree $1$
Conductor $311$
Sign $-0.956 - 0.293i$
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  + (−0.0506 + 0.998i)2-s + (−0.0506 − 0.998i)3-s + (−0.994 − 0.101i)4-s + (0.918 − 0.394i)5-s + 6-s + (0.820 + 0.571i)7-s + (0.151 − 0.988i)8-s + (−0.994 + 0.101i)9-s + (0.347 + 0.937i)10-s + (−0.688 − 0.724i)11-s + (−0.0506 + 0.998i)12-s + (−0.994 + 0.101i)13-s + (−0.612 + 0.790i)14-s + (−0.440 − 0.897i)15-s + (0.979 + 0.201i)16-s + (−0.347 − 0.937i)17-s + ⋯
L(s)  = 1  + (−0.0506 + 0.998i)2-s + (−0.0506 − 0.998i)3-s + (−0.994 − 0.101i)4-s + (0.918 − 0.394i)5-s + 6-s + (0.820 + 0.571i)7-s + (0.151 − 0.988i)8-s + (−0.994 + 0.101i)9-s + (0.347 + 0.937i)10-s + (−0.688 − 0.724i)11-s + (−0.0506 + 0.998i)12-s + (−0.994 + 0.101i)13-s + (−0.612 + 0.790i)14-s + (−0.440 − 0.897i)15-s + (0.979 + 0.201i)16-s + (−0.347 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.956 - 0.293i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ -0.956 - 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02887438738 - 0.1925490906i\)
\(L(\frac12)\) \(\approx\) \(0.02887438738 - 0.1925490906i\)
\(L(1)\) \(\approx\) \(0.8258397467 + 0.05128325049i\)
\(L(1)\) \(\approx\) \(0.8258397467 + 0.05128325049i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.0506 + 0.998i)T \)
3 \( 1 + (-0.0506 - 0.998i)T \)
5 \( 1 + (0.918 - 0.394i)T \)
7 \( 1 + (0.820 + 0.571i)T \)
11 \( 1 + (-0.688 - 0.724i)T \)
13 \( 1 + (-0.994 + 0.101i)T \)
17 \( 1 + (-0.347 - 0.937i)T \)
19 \( 1 + (-0.918 - 0.394i)T \)
23 \( 1 + (-0.151 + 0.988i)T \)
29 \( 1 + (0.612 + 0.790i)T \)
31 \( 1 + (-0.347 + 0.937i)T \)
37 \( 1 + (-0.820 + 0.571i)T \)
41 \( 1 + (0.874 - 0.485i)T \)
43 \( 1 + (-0.820 - 0.571i)T \)
47 \( 1 + (-0.612 - 0.790i)T \)
53 \( 1 + (0.820 + 0.571i)T \)
59 \( 1 + (-0.820 - 0.571i)T \)
61 \( 1 + (-0.918 - 0.394i)T \)
67 \( 1 + (-0.874 - 0.485i)T \)
71 \( 1 + (0.250 + 0.968i)T \)
73 \( 1 + (-0.954 - 0.299i)T \)
79 \( 1 + (-0.874 + 0.485i)T \)
83 \( 1 + (-0.440 + 0.897i)T \)
89 \( 1 + (0.820 - 0.571i)T \)
97 \( 1 + (0.250 + 0.968i)T \)
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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

−25.98904885501130275914790496771, −24.48703045516655225768172970555, −23.18655209144961924595892256608, −22.531981673501811031346098190174, −21.50390172256343999606564363840, −21.10981335093926226757024221708, −20.32066422495347218343431141301, −19.367238803463342510735489101142, −18.02589388520095498506870065520, −17.42920224330979462158771555570, −16.74943220099184920584099007680, −14.87197500369674992607173658432, −14.62797236439309873565629133447, −13.45159495752807435539849662062, −12.44186639929045356633231716214, −11.18616164875437603810394627197, −10.309758645846463203874157929634, −10.08968115435032970117548712925, −8.84835986267331770343092453919, −7.77702723744161512415700764802, −5.99991601679857663428357628463, −4.80712464888656775319588651331, −4.231050215976486681375713676475, −2.70767372250952512526578942764, −1.88339064588825525640376235325, 0.05537543392659131127766050392, 1.51487994298071791221364770582, 2.73611104281921280730810294204, 4.98667905571030809957491059660, 5.3881229750421265807963652615, 6.52036897618976363093094483862, 7.466295142477306515055843318466, 8.52314325846352685348307389498, 9.09651715062744951037906330642, 10.534793259010707122211195247637, 11.94625840679541690334882316268, 12.88811297667110702492462664836, 13.76638769556751753929411141356, 14.32527044133819458286171830410, 15.48163656635024783631776114704, 16.66548814873395930469080415577, 17.49289996132187100462040718068, 18.040985589556389671785408016396, 18.80983313616156529232426758796, 19.926862022349277973188072146268, 21.39687842182958908963577849452, 21.90370596605326013039280205108, 23.22632555771739979071781618010, 24.08839225832473852627112678086, 24.553392843884460953006535346417

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