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Properties

Label 1-916-916.27-r1-0-0
Degree $1$
Conductor $916$
Sign $-0.572 + 0.819i$
Analytic cond. $98.4378$
Root an. cond. $98.4378$
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  + (0.0825 − 0.996i)3-s + (0.245 + 0.969i)5-s + (−0.546 + 0.837i)7-s + (−0.986 − 0.164i)9-s + (0.677 − 0.735i)11-s + (0.245 + 0.969i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (−0.245 + 0.969i)19-s + (0.789 + 0.614i)21-s + (0.401 − 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.245 + 0.969i)27-s + (0.546 − 0.837i)29-s + (0.677 − 0.735i)31-s + ⋯
L(s)  = 1  + (0.0825 − 0.996i)3-s + (0.245 + 0.969i)5-s + (−0.546 + 0.837i)7-s + (−0.986 − 0.164i)9-s + (0.677 − 0.735i)11-s + (0.245 + 0.969i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (−0.245 + 0.969i)19-s + (0.789 + 0.614i)21-s + (0.401 − 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.245 + 0.969i)27-s + (0.546 − 0.837i)29-s + (0.677 − 0.735i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $-0.572 + 0.819i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ -0.572 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5347242976 + 1.025875006i\)
\(L(\frac12)\) \(\approx\) \(0.5347242976 + 1.025875006i\)
\(L(1)\) \(\approx\) \(0.9932683030 + 0.08908079157i\)
\(L(1)\) \(\approx\) \(0.9932683030 + 0.08908079157i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.0825 - 0.996i)T \)
5 \( 1 + (0.245 + 0.969i)T \)
7 \( 1 + (-0.546 + 0.837i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (0.245 + 0.969i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (-0.245 + 0.969i)T \)
23 \( 1 + (0.401 - 0.915i)T \)
29 \( 1 + (0.546 - 0.837i)T \)
31 \( 1 + (0.677 - 0.735i)T \)
37 \( 1 + (0.789 + 0.614i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (-0.789 - 0.614i)T \)
47 \( 1 + (0.986 + 0.164i)T \)
53 \( 1 + (-0.0825 + 0.996i)T \)
59 \( 1 + (-0.789 + 0.614i)T \)
61 \( 1 + (-0.986 - 0.164i)T \)
67 \( 1 + (0.986 + 0.164i)T \)
71 \( 1 + (0.677 + 0.735i)T \)
73 \( 1 + (-0.879 + 0.475i)T \)
79 \( 1 + (-0.546 - 0.837i)T \)
83 \( 1 + (-0.789 + 0.614i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.879 - 0.475i)T \)
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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

−21.397867732236617675817057191604, −20.46898794854598670003741521498, −20.028446291623375754633017659914, −19.55873149556484438232952340218, −17.91446541891356360772229570410, −17.29131622443449853813409359386, −16.65170662117191395123880670876, −15.86484757217000801357050405276, −15.2784489466846152141890180709, −14.1502704773833831658530254799, −13.47156274170386702498657771486, −12.62468242331006563821068325341, −11.67140188969470166602836427018, −10.698266009200930699006284183114, −9.84786153737160502992079832722, −9.34475143440375385739421926710, −8.51229325053662583161550534879, −7.41992849660424436290442793168, −6.39149714867351114685163850355, −5.17925210688419364561802921226, −4.705550932576894348820167418253, −3.69761117275490928495155890804, −2.84546612375334904761596365227, −1.261470081378608877985167017946, −0.254990249471883043928316434, 1.29019337000734909674888444412, 2.27200161277285200523730486425, 3.05243418958595828478533117639, 4.0492509740303848289262530174, 5.840016728084320247774260583159, 6.24675458692618878092892731260, 6.80828479320571704936167406473, 8.064041560786537420417662135883, 8.70972092717050949803519578101, 9.69890009906471677564854759253, 10.72085823966983091136486042105, 11.70850302646173624898282678921, 12.158271130710730841670394434745, 13.245378338095592727183334670096, 13.94558686026963664639368846603, 14.65015979668554952055437162222, 15.386232046391512262261964334883, 16.72799051056326320703417940631, 17.14604043360877757297109827381, 18.43722670831749187905855883239, 18.90291157736585871344094350193, 19.0759846281202924520535165545, 20.23234669284551042217496069507, 21.4607659522403590117060322095, 21.92104473570376892170424604084

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