↑

Properties

Label 1-308-308.195-r1-0-0
Degree $1$
Conductor $308$
Sign $-0.927 - 0.374i$
Analytic cond. $33.0991$
Root an. cond. $33.0991$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (0.309 − 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.309 + 0.951i)39-s + (−0.809 + 0.587i)41-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (0.309 − 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.309 + 0.951i)39-s + (−0.809 + 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.927 - 0.374i$
Analytic conductor: \(33.0991\)
Root analytic conductor: \(33.0991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (1:\ ),\ -0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08209853891 - 0.4226758684i\)
\(L(\frac12)\) \(\approx\) \(0.08209853891 - 0.4226758684i\)
\(L(1)\) \(\approx\) \(0.6888456436 - 0.08910663093i\)
\(L(1)\) \(\approx\) \(0.6888456436 - 0.08910663093i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.309 + 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.36942391247393292408169355948, −24.42815373051765508946200035177, −23.502694016411317605056488929986, −22.84499512894698535569448541476, −22.12648139303637665418414649316, −21.157965324220630629533074645, −19.83028980531787429392832380167, −18.83709082781048892110534041251, −18.35973523014304846460502884644, −17.468725660171916804721974732672, −16.30006848708954106646738130635, −15.66795375012920224520026777019, −14.112578231086044752331035594019, −13.76454517021336551715544682150, −12.07824602000792914038139807994, −11.781524317402574871411380787410, −10.6770549569399154662814050651, −9.80651935398514629083649265735, −8.2177735183448007635705626956, −7.202820696307632111228434484323, −6.52411215627995374980561847415, −5.45112524928738357302285259693, −4.1454277404611614804883614975, −2.774323697256050420737708925483, −1.43449878005283874722686443216, 0.15763224797929290472577388203, 1.31108817604767359643802202480, 3.354130738825350721998529011827, 4.39782725035575932996673975905, 5.34794109212631190359557932006, 6.17581483205833009821537817060, 7.68109676442509668977603141541, 8.692428275903274191635206209615, 9.772933901623870764258818610658, 10.65120803109740200724023219918, 11.74502520660689266509207949883, 12.46116776271730232563068223765, 13.40493964236269748800976277868, 14.865524757522717575335051888296, 15.79490129560283596474354605478, 16.33833788165543001391094651904, 17.40814123451472191100640123171, 17.97728403453326839871557161810, 19.421481994746977793294350514006, 20.332932674269949802024194344754, 21.06280194826112035970391477605, 22.03477033046597996823708487146, 22.8522231292346748344581941456, 23.80181339412356171283653311448, 24.37309708865123250326107180865

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