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Properties

Label 1-311-311.224-r0-0-0
Degree $1$
Conductor $311$
Sign $0.903 - 0.429i$
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  + (−0.0506 − 0.998i)2-s + (−0.0506 + 0.998i)3-s + (−0.994 + 0.101i)4-s + (0.918 + 0.394i)5-s + 6-s + (0.820 − 0.571i)7-s + (0.151 + 0.988i)8-s + (−0.994 − 0.101i)9-s + (0.347 − 0.937i)10-s + (0.688 − 0.724i)11-s + (−0.0506 − 0.998i)12-s + (−0.994 − 0.101i)13-s + (−0.612 − 0.790i)14-s + (−0.440 + 0.897i)15-s + (0.979 − 0.201i)16-s + (0.347 − 0.937i)17-s + ⋯
L(s)  = 1  + (−0.0506 − 0.998i)2-s + (−0.0506 + 0.998i)3-s + (−0.994 + 0.101i)4-s + (0.918 + 0.394i)5-s + 6-s + (0.820 − 0.571i)7-s + (0.151 + 0.988i)8-s + (−0.994 − 0.101i)9-s + (0.347 − 0.937i)10-s + (0.688 − 0.724i)11-s + (−0.0506 − 0.998i)12-s + (−0.994 − 0.101i)13-s + (−0.612 − 0.790i)14-s + (−0.440 + 0.897i)15-s + (0.979 − 0.201i)16-s + (0.347 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.903 - 0.429i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.903 - 0.429i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.338342854 - 0.3021032542i\)
\(L(\frac12)\) \(\approx\) \(1.338342854 - 0.3021032542i\)
\(L(1)\) \(\approx\) \(1.135801021 - 0.2180200026i\)
\(L(1)\) \(\approx\) \(1.135801021 - 0.2180200026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.0506 - 0.998i)T \)
3 \( 1 + (-0.0506 + 0.998i)T \)
5 \( 1 + (0.918 + 0.394i)T \)
7 \( 1 + (0.820 - 0.571i)T \)
11 \( 1 + (0.688 - 0.724i)T \)
13 \( 1 + (-0.994 - 0.101i)T \)
17 \( 1 + (0.347 - 0.937i)T \)
19 \( 1 + (0.918 - 0.394i)T \)
23 \( 1 + (0.151 + 0.988i)T \)
29 \( 1 + (-0.612 + 0.790i)T \)
31 \( 1 + (0.347 + 0.937i)T \)
37 \( 1 + (0.820 + 0.571i)T \)
41 \( 1 + (-0.874 - 0.485i)T \)
43 \( 1 + (0.820 - 0.571i)T \)
47 \( 1 + (-0.612 + 0.790i)T \)
53 \( 1 + (0.820 - 0.571i)T \)
59 \( 1 + (0.820 - 0.571i)T \)
61 \( 1 + (0.918 - 0.394i)T \)
67 \( 1 + (-0.874 + 0.485i)T \)
71 \( 1 + (-0.250 + 0.968i)T \)
73 \( 1 + (-0.954 + 0.299i)T \)
79 \( 1 + (-0.874 - 0.485i)T \)
83 \( 1 + (-0.440 - 0.897i)T \)
89 \( 1 + (0.820 + 0.571i)T \)
97 \( 1 + (-0.250 + 0.968i)T \)
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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

−24.9402349869210386911402070938, −24.62126835732265912636670089035, −23.96720932412736189632092649049, −22.71923607424790424797232257295, −22.059784486812524861444069806602, −20.903829482102063854109715838239, −19.70819111610174212814086663614, −18.608124422945490646587418940857, −17.91769854348465587868726307672, −17.18551456208613376999825568347, −16.64662742050538387773256934638, −14.79498813244176337145300532342, −14.606348667287057146218640422793, −13.472253555180220536906391051950, −12.58551935786013130260244836589, −11.80997726359029968935521995950, −10.036282606320179839142545971516, −9.11578077690661131402733519535, −8.163904410898264083581025463447, −7.30973777252849525358776291097, −6.18372247033515944158355637430, −5.49694070689537554733095646046, −4.43560413815288612590783088964, −2.32657923073757663415017970608, −1.26280741883565496767311186970, 1.222969402574301593773298874401, 2.7052206314090019476254735274, 3.586521850264664973342306569441, 4.91538224524033460361113332036, 5.47547582700495224966906736066, 7.277170510584972220111539780449, 8.73051784324740199928674523437, 9.585864353087580235987146797991, 10.225027418411485539596381744370, 11.23983608368604363547986971174, 11.771313328403209312727629699710, 13.427978803557989900583882675, 14.206383691370028865938585577851, 14.67292796218689317933125329114, 16.33285858530588607581710329437, 17.34164894540840398301566077895, 17.747873339912033882071272190647, 19.03061095808472177765591385629, 20.133626314000835446880860142857, 20.70731315090515920621312150295, 21.74275028336145746458744572819, 22.01018172385787235199075411413, 22.97110605711797018578880431078, 24.19832520578825944121476230272, 25.40821175169461833396037680950

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