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Properties

Label 1-2e5-32.5-r0-0-0
Degree $1$
Conductor $32$
Sign $0.555 + 0.831i$
Analytic cond. $0.148607$
Root an. cond. $0.148607$
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  + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + i·7-s − i·9-s + (−0.707 − 0.707i)11-s + (0.707 − 0.707i)13-s − 15-s − 17-s + (0.707 − 0.707i)19-s + (−0.707 − 0.707i)21-s − i·23-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + i·7-s − i·9-s + (−0.707 − 0.707i)11-s + (0.707 − 0.707i)13-s − 15-s − 17-s + (0.707 − 0.707i)19-s + (−0.707 − 0.707i)21-s − i·23-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(0.148607\)
Root analytic conductor: \(0.148607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 32,\ (0:\ ),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5968585698 + 0.3190275521i\)
\(L(\frac12)\) \(\approx\) \(0.5968585698 + 0.3190275521i\)
\(L(1)\) \(\approx\) \(0.8128533770 + 0.2721024297i\)
\(L(1)\) \(\approx\) \(0.8128533770 + 0.2721024297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 + iT \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + iT \)
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

−36.04287367230733825865025619868, −35.635484288982439277980201306356, −33.60711845578218687389313168416, −33.271760896971241898260960317222, −31.38069032392039881602813044947, −30.06287910870776622604982137380, −28.95924076086137830462881964731, −28.22046986633945729073296017888, −26.3958271755062273420650548793, −24.99402323434118102820059668551, −23.8420931815044262475601627185, −22.88159429767618842563350980403, −21.20792858558038908661694309476, −19.94902534701121500851990746527, −18.21917751697834170267040614454, −17.257161714752641343305979360589, −16.1320222254487730501484775984, −13.74066443950421262090986180363, −12.978060800643753945329949705474, −11.34417060495042936727964131282, −9.81781071538947725499538815108, −7.78989884395503367104593862974, −6.27178192155914422380013490473, −4.68540358759759307267341170842, −1.61796366745704267360678474775, 2.93980914934491876882348312166, 5.26830475631644288871154228503, 6.36900691647196385487732509292, 8.840836184286508683008535274278, 10.35751758728224776296900481828, 11.41447865465362867125541895181, 13.23633554137878311828454913452, 15.022478055371743092833375428495, 16.02997273108670983286583311768, 17.73970088705665045545356340097, 18.52915373087765196114113297930, 20.72886352148557734263381799325, 21.87840323758562610190290265399, 22.5790061497568081238836339054, 24.27706967405060115307228629307, 25.81909098311021865364130611181, 26.87257388047438987945123649392, 28.34579119564014013345931114692, 29.11198174562442421990740925592, 30.6179077939016069215284902426, 32.15736094397393362013552506267, 33.25622908968484106913004639477, 34.33873934094434389032606276883, 35.11950716231660457387726539327, 37.27797253488685759462698003677

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