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Properties

Label 1-229-229.34-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.847 - 0.531i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (−0.969 + 0.245i)2-s + (0.789 − 0.614i)3-s + (0.879 − 0.475i)4-s + (0.401 − 0.915i)5-s + (−0.614 + 0.789i)6-s + (0.996 + 0.0825i)7-s + (−0.735 + 0.677i)8-s + (0.245 − 0.969i)9-s + (−0.164 + 0.986i)10-s + (−0.945 − 0.324i)11-s + (0.401 − 0.915i)12-s + (−0.915 − 0.401i)13-s + (−0.986 + 0.164i)14-s + (−0.245 − 0.969i)15-s + (0.546 − 0.837i)16-s + (−0.401 + 0.915i)17-s + ⋯
L(s)  = 1  + (−0.969 + 0.245i)2-s + (0.789 − 0.614i)3-s + (0.879 − 0.475i)4-s + (0.401 − 0.915i)5-s + (−0.614 + 0.789i)6-s + (0.996 + 0.0825i)7-s + (−0.735 + 0.677i)8-s + (0.245 − 0.969i)9-s + (−0.164 + 0.986i)10-s + (−0.945 − 0.324i)11-s + (0.401 − 0.915i)12-s + (−0.915 − 0.401i)13-s + (−0.986 + 0.164i)14-s + (−0.245 − 0.969i)15-s + (0.546 − 0.837i)16-s + (−0.401 + 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.847 - 0.531i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.847 - 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3448470510 - 1.199251110i\)
\(L(\frac12)\) \(\approx\) \(0.3448470510 - 1.199251110i\)
\(L(1)\) \(\approx\) \(0.8156518261 - 0.4155666596i\)
\(L(1)\) \(\approx\) \(0.8156518261 - 0.4155666596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.969 + 0.245i)T \)
3 \( 1 + (0.789 - 0.614i)T \)
5 \( 1 + (0.401 - 0.915i)T \)
7 \( 1 + (0.996 + 0.0825i)T \)
11 \( 1 + (-0.945 - 0.324i)T \)
13 \( 1 + (-0.915 - 0.401i)T \)
17 \( 1 + (-0.401 + 0.915i)T \)
19 \( 1 + (-0.401 - 0.915i)T \)
23 \( 1 + (-0.164 - 0.986i)T \)
29 \( 1 + (-0.996 - 0.0825i)T \)
31 \( 1 + (-0.324 + 0.945i)T \)
37 \( 1 + (0.546 + 0.837i)T \)
41 \( 1 + (0.969 - 0.245i)T \)
43 \( 1 + (0.546 + 0.837i)T \)
47 \( 1 + (-0.969 - 0.245i)T \)
53 \( 1 + (0.789 - 0.614i)T \)
59 \( 1 + (0.837 + 0.546i)T \)
61 \( 1 + (0.245 - 0.969i)T \)
67 \( 1 + (-0.969 - 0.245i)T \)
71 \( 1 + (-0.945 + 0.324i)T \)
73 \( 1 + (0.735 - 0.677i)T \)
79 \( 1 + (-0.996 + 0.0825i)T \)
83 \( 1 + (0.546 - 0.837i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.677 - 0.735i)T \)
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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

−26.634897779250262243060639990133, −25.85384535140517261698378925713, −25.05552850304801213984906525665, −24.09305215575971567441782086942, −22.49121453952134186541820451792, −21.4117291516035781248159447725, −20.94013076070689712383721126559, −19.99464661812533848649390741560, −18.92920104830344020661865373000, −18.19622572343720096695400673113, −17.28215222191140940978522482045, −16.13071366593393435503162553425, −15.05060340390296836024693862617, −14.469093983026727394991249507359, −13.20789675287864812448043426299, −11.59527127165444032769443930313, −10.75614696889498107618704192978, −9.91854265107097839306811665982, −9.13225346844499813375455806906, −7.69358778219467319972235288848, −7.40047362853785185263439188161, −5.52394653092429328395238046033, −3.99985439644172129862052578359, −2.55401286827627439538135822835, −1.98763650272810497431078053490, 0.45902883128581555000782019746, 1.75455570014505435981161077828, 2.59558440678386727866286531507, 4.73885215354260548519602495914, 5.941796616513511470219113342, 7.31621730092392414883384610064, 8.254208709182664217831574172606, 8.7218780328907387463973175477, 9.88808354479531875753039534036, 11.03877957190088901146590293240, 12.370560125654734812417073552457, 13.21356030048478951447113499956, 14.544488210893518373919960759499, 15.22479695781311381182399486098, 16.47615804354175014992072252135, 17.59975871242822275139304473554, 18.00546327496322175271788205533, 19.2199344564375907332083019779, 20.02959843026343052680811046688, 20.79513341595733323906272248931, 21.543525923484555201946543054189, 23.706298586510878423614394046354, 24.27932936980733252864615492420, 24.70952086898258880233647040084, 25.836460245165852365080291781941

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