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Properties

Label 1-153-153.43-r0-0-0
Degree $1$
Conductor $153$
Sign $0.991 + 0.133i$
Analytic cond. $0.710529$
Root an. cond. $0.710529$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.965 − 0.258i)5-s + (0.965 + 0.258i)7-s + i·8-s + (−0.707 + 0.707i)10-s + (−0.965 − 0.258i)11-s + (0.5 − 0.866i)13-s + (−0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s − i·19-s + (0.258 − 0.965i)20-s + (0.965 − 0.258i)22-s + (0.258 + 0.965i)23-s + (0.866 − 0.5i)25-s + i·26-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.965 − 0.258i)5-s + (0.965 + 0.258i)7-s + i·8-s + (−0.707 + 0.707i)10-s + (−0.965 − 0.258i)11-s + (0.5 − 0.866i)13-s + (−0.965 + 0.258i)14-s + (−0.5 − 0.866i)16-s − i·19-s + (0.258 − 0.965i)20-s + (0.965 − 0.258i)22-s + (0.258 + 0.965i)23-s + (0.866 − 0.5i)25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.991 + 0.133i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.991 + 0.133i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9325356983 + 0.06247835661i\)
\(L(\frac12)\) \(\approx\) \(0.9325356983 + 0.06247835661i\)
\(L(1)\) \(\approx\) \(0.8905729387 + 0.08720896387i\)
\(L(1)\) \(\approx\) \(0.8905729387 + 0.08720896387i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (-0.258 + 0.965i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.258 - 0.965i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.965 + 0.258i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.258 + 0.965i)T \)
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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

−28.176056816508656757430076653368, −26.873370841473353421103181047670, −26.27475224815784678430939341397, −25.28282275384253847420827932484, −24.361872841898386572431840475549, −23.02439657785070323810980377711, −21.58263217197527464348450737244, −21.02227932586354185525387197372, −20.28779847816534255748083120883, −18.61370758235773951065428027130, −18.29524729521518593331755985718, −17.18384784465069358609498264836, −16.36991036746449477603217326926, −14.86327978357153414974542894537, −13.71456422612221088748916586027, −12.61434315675503641205062311012, −11.2493445351544522765790857315, −10.5073421409702867054946634722, −9.4861303734285186022237739584, −8.31983973159368522802528824145, −7.271402350992643488244835027426, −5.89492744053203979903957318518, −4.26160032579825052242964337807, −2.51818630375966484109866723314, −1.54804344737846902505674659889, 1.29928709489507278192029304963, 2.603481295250604919763331575620, 5.14915629142699459092321655188, 5.69098763412014874305261241532, 7.23788170379182798887049727361, 8.35141244758159905185773844773, 9.20791432457852138739649645361, 10.47531263616784081872141785221, 11.20631365912083702838388678747, 12.951379707629084279389737495880, 14.028824836832216504041803788000, 15.14959658559431907521436362314, 16.05995977548175338310290721968, 17.37306650490101236328473573862, 17.87842597338970404115148397819, 18.71607240212989223245238029950, 20.20847227484355383211928421333, 20.90951051725505305792883446231, 21.9804905939846920043074773650, 23.64977656563645647027641606837, 24.205805763245089465668509851653, 25.38362181950637460708758890075, 25.83216013350479549661058602286, 27.14399235008405417809202694098, 27.90235835345200766608868978991

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