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
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (-0.0506 + 0.998i)T \) |
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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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