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Properties

Label 1-311-311.107-r0-0-0
Degree $1$
Conductor $311$
Sign $0.978 - 0.204i$
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  + (0.494 − 0.869i)2-s + (−0.910 − 0.412i)3-s + (−0.511 − 0.859i)4-s + (0.628 + 0.778i)5-s + (−0.809 + 0.587i)6-s + (−0.758 + 0.651i)7-s + (−0.999 + 0.0202i)8-s + (0.659 + 0.752i)9-s + (0.986 − 0.161i)10-s + (−0.994 − 0.101i)11-s + (0.111 + 0.993i)12-s + (0.918 − 0.394i)13-s + (0.191 + 0.981i)14-s + (−0.250 − 0.968i)15-s + (−0.476 + 0.879i)16-s + (0.458 + 0.888i)17-s + ⋯
L(s)  = 1  + (0.494 − 0.869i)2-s + (−0.910 − 0.412i)3-s + (−0.511 − 0.859i)4-s + (0.628 + 0.778i)5-s + (−0.809 + 0.587i)6-s + (−0.758 + 0.651i)7-s + (−0.999 + 0.0202i)8-s + (0.659 + 0.752i)9-s + (0.986 − 0.161i)10-s + (−0.994 − 0.101i)11-s + (0.111 + 0.993i)12-s + (0.918 − 0.394i)13-s + (0.191 + 0.981i)14-s + (−0.250 − 0.968i)15-s + (−0.476 + 0.879i)16-s + (0.458 + 0.888i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.978 - 0.204i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.978 - 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034835951 - 0.1068359753i\)
\(L(\frac12)\) \(\approx\) \(1.034835951 - 0.1068359753i\)
\(L(1)\) \(\approx\) \(0.9309918955 - 0.2831612663i\)
\(L(1)\) \(\approx\) \(0.9309918955 - 0.2831612663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.494 - 0.869i)T \)
3 \( 1 + (-0.910 - 0.412i)T \)
5 \( 1 + (0.628 + 0.778i)T \)
7 \( 1 + (-0.758 + 0.651i)T \)
11 \( 1 + (-0.994 - 0.101i)T \)
13 \( 1 + (0.918 - 0.394i)T \)
17 \( 1 + (0.458 + 0.888i)T \)
19 \( 1 + (0.934 + 0.356i)T \)
23 \( 1 + (-0.289 + 0.957i)T \)
29 \( 1 + (0.992 - 0.121i)T \)
31 \( 1 + (0.986 + 0.161i)T \)
37 \( 1 + (0.230 + 0.972i)T \)
41 \( 1 + (-0.440 - 0.897i)T \)
43 \( 1 + (-0.853 - 0.520i)T \)
47 \( 1 + (-0.874 - 0.485i)T \)
53 \( 1 + (0.996 - 0.0809i)T \)
59 \( 1 + (0.385 + 0.922i)T \)
61 \( 1 + (-0.0506 + 0.998i)T \)
67 \( 1 + (0.884 - 0.467i)T \)
71 \( 1 + (0.970 + 0.240i)T \)
73 \( 1 + (-0.784 + 0.620i)T \)
79 \( 1 + (-0.989 + 0.141i)T \)
83 \( 1 + (-0.250 + 0.968i)T \)
89 \( 1 + (-0.758 - 0.651i)T \)
97 \( 1 + (0.0708 + 0.997i)T \)
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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

−25.140625500237287485083243256602, −24.262754980511058215690536323977, −23.27377722411991624216355118703, −22.96321567907167508420968078360, −21.81334309209326019018738860341, −21.04447866036696419905786489006, −20.339094358873939527244264040415, −18.43323625437215492070500532910, −17.83102912846773195334147601905, −16.788476922313013637935390065412, −16.10685637930312428670001167104, −15.811581951202664412437642683575, −14.15101565533608356838994332571, −13.32594545246746234702987375572, −12.63853959896505349679318253410, −11.57639990835696188214735205935, −10.149771078323610515354302499183, −9.45313219448585875590451025895, −8.19050013653737007778871771206, −6.870789882696755565348960562212, −6.10433753383377991354470108899, −5.12026153446465734728686755418, −4.411320780606677077347726961620, −3.08038895611558503181311067213, −0.71942181478943428078389317438, 1.34244370127084406887390217909, 2.59327205277290439623278694954, 3.558094961082099575271234736022, 5.35901031661008493740816716485, 5.80265904814697183178792354743, 6.72546103241013246455470709273, 8.32472598269999398159918764275, 10.06446622337299723141106143691, 10.21592099037825754402384148791, 11.42788872311381146871704700463, 12.22170074485115824745694334940, 13.28618195016776610971282571713, 13.66969804440176282302092096210, 15.20737735509594843068722108882, 15.93928123442147648637667083202, 17.43600587341726224876650622335, 18.38531897013625434529701928800, 18.64696703606303170745282904493, 19.69615323676143552709056224168, 21.1867808938182830704064035634, 21.61732251869688022054613137245, 22.61416003435028484687809680889, 23.06652824770727804957723869617, 23.974969921746217552609806932917, 25.15285391834986556297145394206

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