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Properties

Label 1-152-152.11-r1-0-0
Degree $1$
Conductor $152$
Sign $-0.305 - 0.952i$
Analytic cond. $16.3346$
Root an. cond. $16.3346$
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  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s + 11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s + 11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6546393926 - 0.8978072312i\)
\(L(\frac12)\) \(\approx\) \(0.6546393926 - 0.8978072312i\)
\(L(1)\) \(\approx\) \(0.8404346409 - 0.2699053707i\)
\(L(1)\) \(\approx\) \(0.8404346409 - 0.2699053707i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−28.190233849836519985119902007506, −27.27099473190211334392124129419, −26.0677009997664327295389456771, −25.37158634238380915586504729349, −24.05332246268481185163285162085, −23.11674064409503310492322030938, −21.948887620679708352192514290002, −21.45914270356078081830928827651, −20.22828527280959979975628266993, −19.400881083340813757670435597436, −17.817994710858713387236800966004, −16.79278498381389134423974663422, −16.33522300127354192024986990512, −15.19751818734286401585872609526, −13.876210897634160209824950413607, −12.7327877700280747706532352546, −11.72306955300970982763877694166, −10.48166027773724319977399367863, −9.2902766297732917579655450400, −8.90030275353025450796485732112, −6.66200777806023815913407706983, −5.83331846201007071478383458744, −4.50561691194787151793307987885, −3.491800122126944042220834625768, −1.3650927250625692420893540971, 0.473583690225001973865932685105, 2.20208646241402772319265226454, 3.44557847114345774774327111419, 5.48081472186279465291754994642, 6.54592414214426878214092212596, 7.079726973088655034888089480183, 8.742920710928747263199605413554, 10.08293854605747487592896982653, 11.08918677862201887280474304211, 12.22134099107134383717519706368, 13.29851105553702268767248099091, 14.056908632268930651648144186302, 15.42267994588746653568416592711, 16.69585672575468443432119137095, 17.62770844808576588795315136252, 18.54238525076061990450483573321, 19.299100640154004212433587226741, 20.40333888255048125983546468732, 22.15867373591211527990659044358, 22.49585464017863927510466997751, 23.35039009799455609393005912090, 24.90776311915742593976435419913, 25.23750723109587970063652019848, 26.40299733358719653884143581911, 27.583016462216010341620369988910

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