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Properties

Label 1-311-311.121-r0-0-0
Degree $1$
Conductor $311$
Sign $0.623 - 0.782i$
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  + (0.528 + 0.848i)2-s + (0.528 − 0.848i)3-s + (−0.440 + 0.897i)4-s + (−0.250 − 0.968i)5-s + 6-s + (0.918 − 0.394i)7-s + (−0.994 + 0.101i)8-s + (−0.440 − 0.897i)9-s + (0.688 − 0.724i)10-s + (−0.874 − 0.485i)11-s + (0.528 + 0.848i)12-s + (−0.440 − 0.897i)13-s + (0.820 + 0.571i)14-s + (−0.954 − 0.299i)15-s + (−0.612 − 0.790i)16-s + (0.688 − 0.724i)17-s + ⋯
L(s)  = 1  + (0.528 + 0.848i)2-s + (0.528 − 0.848i)3-s + (−0.440 + 0.897i)4-s + (−0.250 − 0.968i)5-s + 6-s + (0.918 − 0.394i)7-s + (−0.994 + 0.101i)8-s + (−0.440 − 0.897i)9-s + (0.688 − 0.724i)10-s + (−0.874 − 0.485i)11-s + (0.528 + 0.848i)12-s + (−0.440 − 0.897i)13-s + (0.820 + 0.571i)14-s + (−0.954 − 0.299i)15-s + (−0.612 − 0.790i)16-s + (0.688 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.623 - 0.782i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.623 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.485511980 - 0.7158594936i\)
\(L(\frac12)\) \(\approx\) \(1.485511980 - 0.7158594936i\)
\(L(1)\) \(\approx\) \(1.408306443 - 0.1655556023i\)
\(L(1)\) \(\approx\) \(1.408306443 - 0.1655556023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.528 + 0.848i)T \)
3 \( 1 + (0.528 - 0.848i)T \)
5 \( 1 + (-0.250 - 0.968i)T \)
7 \( 1 + (0.918 - 0.394i)T \)
11 \( 1 + (-0.874 - 0.485i)T \)
13 \( 1 + (-0.440 - 0.897i)T \)
17 \( 1 + (0.688 - 0.724i)T \)
19 \( 1 + (-0.250 + 0.968i)T \)
23 \( 1 + (-0.994 + 0.101i)T \)
29 \( 1 + (0.820 - 0.571i)T \)
31 \( 1 + (0.688 + 0.724i)T \)
37 \( 1 + (0.918 + 0.394i)T \)
41 \( 1 + (-0.758 + 0.651i)T \)
43 \( 1 + (0.918 - 0.394i)T \)
47 \( 1 + (0.820 - 0.571i)T \)
53 \( 1 + (0.918 - 0.394i)T \)
59 \( 1 + (0.918 - 0.394i)T \)
61 \( 1 + (-0.250 + 0.968i)T \)
67 \( 1 + (-0.758 - 0.651i)T \)
71 \( 1 + (0.347 + 0.937i)T \)
73 \( 1 + (0.979 - 0.201i)T \)
79 \( 1 + (-0.758 + 0.651i)T \)
83 \( 1 + (-0.954 + 0.299i)T \)
89 \( 1 + (0.918 + 0.394i)T \)
97 \( 1 + (0.347 + 0.937i)T \)
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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

−25.63437684473161364743487464757, −24.148400264424577907800632896477, −23.47341440974441134704828537745, −22.34533550394834962839181366168, −21.60410883511043901977967033224, −21.16639351183076314449535516842, −20.112902842045012967885595164370, −19.258885957382452640124193853693, −18.462982386225399104176358733550, −17.4608669304967202435706570910, −15.80270626584014755306879468054, −15.047951560950016541767786994176, −14.44134151549101126023450776502, −13.70094761266066735858131843302, −12.26289557031589285951193256116, −11.33484883108027388481364373315, −10.554811344138091541492400112909, −9.83437955189548247304856846407, −8.65268825175059283398938357337, −7.54664456982142908341510913991, −5.93000954921171838290244665959, −4.75533773209805418540547738355, −4.050123791913799579671555598905, −2.70850243717132102947922050132, −2.12420915704641356428190816427, 0.856527498450456283150574644827, 2.58056515972656551160129095383, 3.87564315872533187647188226395, 5.07266310993289987951896331793, 5.86923712772451957249118078791, 7.37668408904816147005968559169, 8.0956163705383808384023399115, 8.41931286655044828768347427826, 9.9766733760207516913242336189, 11.79403623513893432796641480378, 12.37297951867840190580550765335, 13.39601057449779624510579155952, 13.99265624519862810552625383381, 14.968512433620315738201423212052, 15.951842072930925144233062328258, 16.93708522840249130567092189116, 17.78534556559803988701262674944, 18.554727844912847962528866460672, 19.92491134756157926411504682524, 20.74002909805267602009403714155, 21.33159584359955994025125664653, 22.95489666390415915216456456015, 23.58094216305749187328505106055, 24.25127342912147115914088775142, 24.93135817366857965144088478267

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