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Properties

Label 1-2e5-32.11-r1-0-0
Degree $1$
Conductor $32$
Sign $-0.555 - 0.831i$
Analytic cond. $3.43887$
Root an. cond. $3.43887$
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

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

Invariants

Degree: \(1\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(3.43887\)
Root analytic conductor: \(3.43887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 32,\ (1:\ ),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2157838221 - 0.4037031366i\)
\(L(\frac12)\) \(\approx\) \(0.2157838221 - 0.4037031366i\)
\(L(1)\) \(\approx\) \(0.5895674212 - 0.1172722514i\)
\(L(1)\) \(\approx\) \(0.5895674212 - 0.1172722514i\)

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

−36.563320831480398187828733150276, −35.11018580848094682067979262108, −34.595983483177342598268965004987, −33.42615366434263286749973258700, −31.39405175047235851176092492747, −30.71386947741491043347034624890, −29.27947007484902521818795674033, −28.2262756654867517525883862310, −26.97053742034207476158609514880, −25.3268854178902836268812067947, −24.15325091339672301916403862563, −22.83135187919236460034550904823, −22.07061394708290993310184347701, −19.97480755039527644488342556301, −18.579847968191037089237283116568, −17.88874729907091804151002217436, −16.01486636107363878259656233839, −14.78330839875493402363799741298, −12.78239739432504252037980505139, −11.80290245042927264818206101295, −10.401582824759769374713253267141, −8.08103896471699337191156739253, −6.77857064340597392951572332309, −5.12118384634175354038228292470, −2.52335569027335173359802788495, 0.335997693309994745288286596145, 3.90396140808436788368909584674, 5.15648920368548608009817582684, 7.237171006330304127500147324182, 9.11566793887460192425536653720, 10.7279912516477488942001281778, 11.85753765983173802371176480683, 13.54817089184996679921492779725, 15.51481202317953046138317638911, 16.4502156222690426019925636176, 17.55455700881077178097288024770, 19.54007392555487010952974071140, 20.71492265311507038096430588898, 22.00774425634167274349368024193, 23.487689357666505901027686275079, 24.13909475814481445734606320200, 26.48108629740884827349656064020, 27.08141175091534323190065024537, 28.508123922038415751517514977483, 29.38119164680306878354992438225, 31.2178771080785969480708569384, 32.34719867997491334363745730981, 33.40493274860043366950913642966, 34.603647099619579339330344930451, 35.76899755798925873865168627354

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