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Properties

Label 1-129-129.122-r1-0-0
Degree $1$
Conductor $129$
Sign $-0.953 + 0.300i$
Analytic cond. $13.8629$
Root an. cond. $13.8629$
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  − 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + 22-s + (0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ -0.953 + 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03615844924 - 0.2353308209i\)
\(L(\frac12)\) \(\approx\) \(-0.03615844924 - 0.2353308209i\)
\(L(1)\) \(\approx\) \(0.5276858188 - 0.1660780587i\)
\(L(1)\) \(\approx\) \(0.5276858188 - 0.1660780587i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−29.03522501170501154318801832593, −28.1206191597319356189288590001, −26.91887419343223544510267905898, −26.03275190015927267541419261951, −25.459050700864905606186391924099, −24.36266241937688865634542437938, −23.05478449571134087948160693315, −21.675556679584680499956403045489, −21.11549870160749406544526387898, −19.5271935121401541522211418901, −18.75357414612502488053500565658, −18.06882637372511528184852746412, −16.93372474774014827292626921708, −15.716629583512535393340716043183, −14.95966784956478843441987548062, −13.487030153171980382579819011998, −12.023222715902566529334051352857, −11.01079013087255452253627601718, −9.83612964691918015328026085169, −9.125765724152142816939878514141, −7.592253669142676657865294745376, −6.61916273676916405221723988818, −5.446072458204920963818193209468, −2.97350324447509995713770049471, −2.15828803848980640401342280455, 0.125054879542561778123640950422, 1.52203223969758141817802424814, 3.1779818811267883556010548438, 5.12039918182058396023766785916, 6.433589354955794626806573639730, 7.79424631159557126256996380276, 8.66782886484379749915427296856, 10.20665963609649896573748979328, 10.399322439234018799425631419386, 12.36427278089702166346240530692, 13.05258651899887887377996368095, 14.67074151800103840542231497709, 16.02891302910166105548586043449, 16.79503541655209588772062903637, 17.57108692198216683797116319183, 18.75132963252247693975786554184, 19.90995920182994270623197051898, 20.588190785208071721792583801759, 21.53087613826015052279913602942, 23.21496014612266136570012206403, 24.14144069767960218095294346590, 25.25219873663266508981118260516, 25.9330759050567959949957519565, 27.03186462493784746883127453528, 27.90969850984572625919240164805

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