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Properties

Label 1-2e5-32.29-r0-0-0
Degree $1$
Conductor $32$
Sign $0.831 + 0.555i$
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

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

Invariants

Degree: \(1\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.831 + 0.555i$
Analytic conductor: \(0.148607\)
Root analytic conductor: \(0.148607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 32,\ (0:\ ),\ 0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7969684494 + 0.2417577360i\)
\(L(\frac12)\) \(\approx\) \(0.7969684494 + 0.2417577360i\)
\(L(1)\) \(\approx\) \(1.012975691 + 0.2138733330i\)
\(L(1)\) \(\approx\) \(1.012975691 + 0.2138733330i\)

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

−36.20350179842608913420178686591, −35.55755422315961523221760036174, −34.34387020947609737957986842652, −32.457015952033051194784995940441, −31.4171103920484692173867853411, −30.729521429612034795059730910646, −29.08440065835051747602932838639, −27.92052182446549550368239295172, −26.48799908497637749150262788891, −24.9254448069711538350572794677, −24.38551129533217145706266670991, −22.84467694102314894095181619414, −21.08941396553547631988238970952, −19.77656624005411640344479335117, −18.93307526224527398048631155899, −17.35029326376551831968410870985, −15.58119910909992541977281714715, −14.431147901555861288032844235250, −12.625353747572036786697191452355, −11.91882458571795196144177639340, −9.27675058392445512892036410164, −8.271598278038628119782902138912, −6.6398393015379132960083902410, −4.3448045258246410580413065085, −2.15208420554387225095307454879, 3.09801966453684513256708219014, 4.40684683985788739995057722719, 6.99507831886350090201430875271, 8.43358886805945453495496241207, 10.17677003656408939283643058093, 11.2847837848574952180905229071, 13.52345870218766479179208484079, 14.715097463424444363546900524901, 15.83100391884839530533681294731, 17.34451774957819191792199275591, 19.41809644044323659467711975882, 19.938273892243425342929671918579, 21.65139446261515015520355675173, 22.72955031924795867396962783889, 24.25450334933837620645143290062, 25.85720764094111850630162270677, 26.90220696742894999749910667233, 27.51811555427233192703713875456, 29.65350405767168771697205355224, 30.63389287377528789023311073188, 31.913078752825063389185876766123, 32.93388397584925377977585583806, 34.165682560498602394229851144830, 35.53254479210435668075849244663, 36.95187417213025362145418580185

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