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Properties

Label 1-311-311.243-r0-0-0
Degree $1$
Conductor $311$
Sign $-0.694 - 0.719i$
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  + (0.347 + 0.937i)2-s + (0.347 − 0.937i)3-s + (−0.758 + 0.651i)4-s + (−0.954 + 0.299i)5-s + 6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.758 − 0.651i)9-s + (−0.612 − 0.790i)10-s + (0.820 + 0.571i)11-s + (0.347 + 0.937i)12-s + (−0.758 − 0.651i)13-s + (−0.994 − 0.101i)14-s + (−0.0506 + 0.998i)15-s + (0.151 − 0.988i)16-s + (−0.612 − 0.790i)17-s + ⋯
L(s)  = 1  + (0.347 + 0.937i)2-s + (0.347 − 0.937i)3-s + (−0.758 + 0.651i)4-s + (−0.954 + 0.299i)5-s + 6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.758 − 0.651i)9-s + (−0.612 − 0.790i)10-s + (0.820 + 0.571i)11-s + (0.347 + 0.937i)12-s + (−0.758 − 0.651i)13-s + (−0.994 − 0.101i)14-s + (−0.0506 + 0.998i)15-s + (0.151 − 0.988i)16-s + (−0.612 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.694 - 0.719i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ -0.694 - 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003411141796 + 0.008039688038i\)
\(L(\frac12)\) \(\approx\) \(0.003411141796 + 0.008039688038i\)
\(L(1)\) \(\approx\) \(0.6790363247 + 0.2231226598i\)
\(L(1)\) \(\approx\) \(0.6790363247 + 0.2231226598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.347 + 0.937i)T \)
3 \( 1 + (0.347 - 0.937i)T \)
5 \( 1 + (-0.954 + 0.299i)T \)
7 \( 1 + (-0.440 + 0.897i)T \)
11 \( 1 + (0.820 + 0.571i)T \)
13 \( 1 + (-0.758 - 0.651i)T \)
17 \( 1 + (-0.612 - 0.790i)T \)
19 \( 1 + (-0.954 - 0.299i)T \)
23 \( 1 + (-0.874 - 0.485i)T \)
29 \( 1 + (-0.994 + 0.101i)T \)
31 \( 1 + (-0.612 + 0.790i)T \)
37 \( 1 + (-0.440 - 0.897i)T \)
41 \( 1 + (0.918 + 0.394i)T \)
43 \( 1 + (-0.440 + 0.897i)T \)
47 \( 1 + (-0.994 + 0.101i)T \)
53 \( 1 + (-0.440 + 0.897i)T \)
59 \( 1 + (-0.440 + 0.897i)T \)
61 \( 1 + (-0.954 - 0.299i)T \)
67 \( 1 + (0.918 - 0.394i)T \)
71 \( 1 + (0.979 + 0.201i)T \)
73 \( 1 + (0.528 + 0.848i)T \)
79 \( 1 + (0.918 + 0.394i)T \)
83 \( 1 + (-0.0506 - 0.998i)T \)
89 \( 1 + (-0.440 - 0.897i)T \)
97 \( 1 + (0.979 + 0.201i)T \)
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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

−24.287375359191876185498692370425, −23.68044758942864012461430799657, −22.54515787335114692421076536190, −22.04205408199179118592734351795, −21.02177617565517539149942612556, −20.10807779195745970608474308255, −19.56261808371582985055257316070, −19.02515267039264118188761618223, −17.1487265833970321992988457827, −16.58890087871037931266733833306, −15.33653704986641076344120489558, −14.559335060420730487641426114533, −13.674128348453298089251883519302, −12.60303113910376686361483043827, −11.48793143996260403499535104581, −10.86070189319221397989966489053, −9.803714873407300404360253453201, −8.99093339179128225214392729859, −7.97151024306590132163716221729, −6.35340087506788102115523132155, −4.8629136602179957652636524989, −3.7992858144552870121960674272, −3.73987297049815117694839144059, −1.964774358055359462119300993134, −0.00454942952756584255202070833, 2.392379653959517415492080296749, 3.451953804913889027704279802, 4.70466037579015014657181742935, 6.08933056422323689994116311212, 6.90042851017512035287599281232, 7.65850863269451474235797080784, 8.643150213152931489102439752714, 9.444982058582311173490253419480, 11.39989269863587319098581801238, 12.49918666566374204792413691044, 12.66600671290630193585319954230, 14.19074188869981663749095795608, 14.867601713220645890269168382737, 15.54248883521800879686903558771, 16.65207217134154906315603280366, 17.8198965034221544071814423879, 18.406631103821441030412334883360, 19.472500581144739551656110235455, 20.08669440702751199925173061737, 21.71809558466128547511266805754, 22.69803874796105159117771624718, 23.02902758713493831768952463455, 24.38439903014865095655701500981, 24.64354089970969878596446976563, 25.63679131907453805581533679601

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