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Properties

Label 1-311-311.178-r0-0-0
Degree $1$
Conductor $311$
Sign $0.970 + 0.241i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (−0.703 − 0.710i)2-s + (−0.893 − 0.449i)3-s + (−0.0101 + 0.999i)4-s + (−0.784 + 0.620i)5-s + (0.309 + 0.951i)6-s + (−0.250 − 0.968i)7-s + (0.717 − 0.696i)8-s + (0.595 + 0.803i)9-s + (0.992 + 0.121i)10-s + (−0.758 − 0.651i)11-s + (0.458 − 0.888i)12-s + (−0.954 − 0.299i)13-s + (−0.511 + 0.859i)14-s + (0.979 − 0.201i)15-s + (−0.999 − 0.0202i)16-s + (−0.731 − 0.681i)17-s + ⋯
L(s)  = 1  + (−0.703 − 0.710i)2-s + (−0.893 − 0.449i)3-s + (−0.0101 + 0.999i)4-s + (−0.784 + 0.620i)5-s + (0.309 + 0.951i)6-s + (−0.250 − 0.968i)7-s + (0.717 − 0.696i)8-s + (0.595 + 0.803i)9-s + (0.992 + 0.121i)10-s + (−0.758 − 0.651i)11-s + (0.458 − 0.888i)12-s + (−0.954 − 0.299i)13-s + (−0.511 + 0.859i)14-s + (0.979 − 0.201i)15-s + (−0.999 − 0.0202i)16-s + (−0.731 − 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.970 + 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2871070718 + 0.03512063171i\)
\(L(\frac12)\) \(\approx\) \(0.2871070718 + 0.03512063171i\)
\(L(1)\) \(\approx\) \(0.3903721420 - 0.1279038078i\)
\(L(1)\) \(\approx\) \(0.3903721420 - 0.1279038078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.703 - 0.710i)T \)
3 \( 1 + (-0.893 - 0.449i)T \)
5 \( 1 + (-0.784 + 0.620i)T \)
7 \( 1 + (-0.250 - 0.968i)T \)
11 \( 1 + (-0.758 - 0.651i)T \)
13 \( 1 + (-0.954 - 0.299i)T \)
17 \( 1 + (-0.731 - 0.681i)T \)
19 \( 1 + (0.270 + 0.962i)T \)
23 \( 1 + (-0.171 + 0.985i)T \)
29 \( 1 + (-0.0910 + 0.995i)T \)
31 \( 1 + (0.992 - 0.121i)T \)
37 \( 1 + (0.843 + 0.537i)T \)
41 \( 1 + (-0.0506 - 0.998i)T \)
43 \( 1 + (-0.366 + 0.930i)T \)
47 \( 1 + (0.918 + 0.394i)T \)
53 \( 1 + (-0.998 - 0.0607i)T \)
59 \( 1 + (0.771 + 0.635i)T \)
61 \( 1 + (0.347 - 0.937i)T \)
67 \( 1 + (0.934 + 0.356i)T \)
71 \( 1 + (-0.983 + 0.181i)T \)
73 \( 1 + (0.960 - 0.279i)T \)
79 \( 1 + (0.628 + 0.778i)T \)
83 \( 1 + (0.979 + 0.201i)T \)
89 \( 1 + (-0.250 + 0.968i)T \)
97 \( 1 + (0.902 + 0.431i)T \)
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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

−25.072565538757462490726084735025, −24.25856945605178818279245208277, −23.63947646272901416186446263770, −22.70099524662751116679302023384, −21.84973203044899392961207614062, −20.61435014491181437833731731097, −19.62207588847957846200012918891, −18.72805552414150677575913579614, −17.77001118972749147776241118736, −17.03176747060743917001263568550, −16.07224230783517558201878534310, −15.43292988083617623178431491783, −14.93007508177684603414513656138, −13.04532663712384192758177447641, −12.11243357772010514496466471770, −11.24316721718271758648106265525, −10.09200754517246185527860740181, −9.26582348714815918411755361425, −8.30177780403633952617493398419, −7.160693074976354533439889887246, −6.15666322391228945101333010233, −5.01579979307767614170643888308, −4.47121783466572577113185094580, −2.28857119927678176384721172170, −0.36317780594895974769408707729, 0.88503679628898914088493897090, 2.562458530673995116175509094661, 3.69672016595565258773760733482, 4.915395781443203023634864349819, 6.57475565322646006004352915641, 7.49346444469365890046092501081, 7.984619849865743703253403872002, 9.77816410442218331292407627570, 10.57738881282198699941991651243, 11.24680064425693230637198506767, 12.07605137023501950892933106703, 13.0412310730800439014150100515, 13.9748550612105681620701031325, 15.72221528306355969654937457830, 16.38780756613780924614551757669, 17.33911872127113411352773095531, 18.15582853758959267445474989989, 18.9669311774615873484606217331, 19.65817917187319220937041507470, 20.57396065334340271124587359505, 21.88735364246613338579003495242, 22.51384392133254529018836696519, 23.35689957462799817196355636363, 24.217733914399773582120733846, 25.43172295194185310445284379263

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