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Properties

Label 1-311-311.7-r0-0-0
Degree $1$
Conductor $311$
Sign $0.997 - 0.0770i$
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  + (0.918 − 0.394i)2-s + (0.918 + 0.394i)3-s + (0.688 − 0.724i)4-s + (−0.994 − 0.101i)5-s + 6-s + (0.151 + 0.988i)7-s + (0.347 − 0.937i)8-s + (0.688 + 0.724i)9-s + (−0.954 + 0.299i)10-s + (0.979 − 0.201i)11-s + (0.918 − 0.394i)12-s + (0.688 + 0.724i)13-s + (0.528 + 0.848i)14-s + (−0.874 − 0.485i)15-s + (−0.0506 − 0.998i)16-s + (−0.954 + 0.299i)17-s + ⋯
L(s)  = 1  + (0.918 − 0.394i)2-s + (0.918 + 0.394i)3-s + (0.688 − 0.724i)4-s + (−0.994 − 0.101i)5-s + 6-s + (0.151 + 0.988i)7-s + (0.347 − 0.937i)8-s + (0.688 + 0.724i)9-s + (−0.954 + 0.299i)10-s + (0.979 − 0.201i)11-s + (0.918 − 0.394i)12-s + (0.688 + 0.724i)13-s + (0.528 + 0.848i)14-s + (−0.874 − 0.485i)15-s + (−0.0506 − 0.998i)16-s + (−0.954 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.997 - 0.0770i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.997 - 0.0770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.660305544 - 0.1026677424i\)
\(L(\frac12)\) \(\approx\) \(2.660305544 - 0.1026677424i\)
\(L(1)\) \(\approx\) \(2.094459370 - 0.1286660314i\)
\(L(1)\) \(\approx\) \(2.094459370 - 0.1286660314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.918 - 0.394i)T \)
3 \( 1 + (0.918 + 0.394i)T \)
5 \( 1 + (-0.994 - 0.101i)T \)
7 \( 1 + (0.151 + 0.988i)T \)
11 \( 1 + (0.979 - 0.201i)T \)
13 \( 1 + (0.688 + 0.724i)T \)
17 \( 1 + (-0.954 + 0.299i)T \)
19 \( 1 + (-0.994 + 0.101i)T \)
23 \( 1 + (0.347 - 0.937i)T \)
29 \( 1 + (0.528 - 0.848i)T \)
31 \( 1 + (-0.954 - 0.299i)T \)
37 \( 1 + (0.151 - 0.988i)T \)
41 \( 1 + (-0.612 - 0.790i)T \)
43 \( 1 + (0.151 + 0.988i)T \)
47 \( 1 + (0.528 - 0.848i)T \)
53 \( 1 + (0.151 + 0.988i)T \)
59 \( 1 + (0.151 + 0.988i)T \)
61 \( 1 + (-0.994 + 0.101i)T \)
67 \( 1 + (-0.612 + 0.790i)T \)
71 \( 1 + (-0.440 + 0.897i)T \)
73 \( 1 + (-0.758 - 0.651i)T \)
79 \( 1 + (-0.612 - 0.790i)T \)
83 \( 1 + (-0.874 + 0.485i)T \)
89 \( 1 + (0.151 - 0.988i)T \)
97 \( 1 + (-0.440 + 0.897i)T \)
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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

−25.298207543455834125710685633842, −24.11445901976321666685479017524, −23.62376159892579802011507862025, −22.83497756824530786339964738158, −21.78153637898751951952191357666, −20.50841506389223873727318739284, −20.071204509568934187184376343355, −19.38251996067951372674946604529, −17.95318481954435733605234725076, −16.928087433782742653820951844313, −15.782152810963517404249713855, −15.084357715254927891860121459678, −14.30087449277261999292222217852, −13.39442058701815998515311802977, −12.668016192755907689644571629184, −11.55125412848817745974630618201, −10.65808306827712447435203154237, −8.893005788090502761839120076391, −8.04931013565122811916093168237, −7.12544435465825277616485142999, −6.5433394912123825340341804194, −4.66041831825417322783937559960, −3.834197254667026704939010549799, −3.13546390932866415348584234421, −1.4969731169516295633306531661, 1.75438411486411341521529072635, 2.801109511188100170255506985212, 4.07331596966843145044836447522, 4.38696123763224396554092960224, 6.01577604071470134526123550206, 7.0960605097559731907416046157, 8.59063754218573036384005566505, 9.05472695139487183465212912319, 10.632875869897596682189890050808, 11.44657837281123612030056558124, 12.35426822307695561356033772658, 13.29403778926075993961054647688, 14.44842694215150153828689413312, 15.00710194384347462556105657446, 15.75240171950139569738020147323, 16.59798664200862110890994845770, 18.59839827164743029601999364135, 19.24010398322880302013950160852, 19.89863710906189877820874505710, 20.85138498674813409710883004219, 21.62294194754724987570964749442, 22.35170978463933308091532400808, 23.408764751259875683979867198100, 24.44043584078959957394454581595, 24.90425029228442475923103869279

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