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Properties

Label 1-2032-2032.1133-r0-0-0
Degree $1$
Conductor $2032$
Sign $0.918 + 0.394i$
Analytic cond. $9.43656$
Root an. cond. $9.43656$
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  + (0.997 + 0.0747i)3-s + (0.974 + 0.222i)5-s + (0.733 + 0.680i)7-s + (0.988 + 0.149i)9-s + (−0.930 − 0.365i)11-s + (0.680 − 0.733i)13-s + (0.955 + 0.294i)15-s + (0.955 − 0.294i)17-s + i·19-s + (0.680 + 0.733i)21-s + (−0.365 − 0.930i)23-s + (0.900 + 0.433i)25-s + (0.974 + 0.222i)27-s + (−0.563 + 0.826i)29-s + (0.955 + 0.294i)31-s + ⋯
L(s)  = 1  + (0.997 + 0.0747i)3-s + (0.974 + 0.222i)5-s + (0.733 + 0.680i)7-s + (0.988 + 0.149i)9-s + (−0.930 − 0.365i)11-s + (0.680 − 0.733i)13-s + (0.955 + 0.294i)15-s + (0.955 − 0.294i)17-s + i·19-s + (0.680 + 0.733i)21-s + (−0.365 − 0.930i)23-s + (0.900 + 0.433i)25-s + (0.974 + 0.222i)27-s + (−0.563 + 0.826i)29-s + (0.955 + 0.294i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2032\)    =    \(2^{4} \cdot 127\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(9.43656\)
Root analytic conductor: \(9.43656\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2032} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2032,\ (0:\ ),\ 0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.304399773 + 0.6791610118i\)
\(L(\frac12)\) \(\approx\) \(3.304399773 + 0.6791610118i\)
\(L(1)\) \(\approx\) \(1.935744144 + 0.2177689851i\)
\(L(1)\) \(\approx\) \(1.935744144 + 0.2177689851i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
127 \( 1 \)
good3 \( 1 + (0.997 + 0.0747i)T \)
5 \( 1 + (0.974 + 0.222i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (-0.930 - 0.365i)T \)
13 \( 1 + (0.680 - 0.733i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + iT \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (-0.563 + 0.826i)T \)
31 \( 1 + (0.955 + 0.294i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.955 + 0.294i)T \)
43 \( 1 + (0.563 - 0.826i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.149 + 0.988i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.433 - 0.900i)T \)
67 \( 1 + (0.997 + 0.0747i)T \)
71 \( 1 + (-0.365 - 0.930i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 + (0.365 + 0.930i)T \)
83 \( 1 + (0.149 - 0.988i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (0.826 - 0.563i)T \)
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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

−20.006282547883683245491809769462, −19.11906461157385983921237674665, −18.4696741331539004257060723639, −17.65072908830699884241128498272, −17.189992605632675634908760926368, −16.10603553399069823538590749027, −15.46624136707348836936698565137, −14.53677499869391886856314461550, −13.95691613265665408343487496158, −13.40831581461293200231918675542, −12.89283862871485275525473318954, −11.76517275670981768684279916583, −10.81059519446490073633742383259, −10.03704065078972971508791367275, −9.50221432077119860362049883170, −8.61499192506174593522168527372, −7.89840437094874330756519247964, −7.24741836218912569675555250929, −6.29958356034460160029793193112, −5.27089389960340125123533046237, −4.51848192368847639108496712460, −3.651840169388185081401853068248, −2.61326261607429779147391896462, −1.820371898088425178893353771231, −1.15342059269444177161986906795, 1.28502214359080303053832761054, 2.00497718337317434964952378012, 2.89642642256345799528927777517, 3.43928623694088019772135754409, 4.81554183365066732869588255924, 5.46878726079228988362482474528, 6.20961373570693754713960382754, 7.35817878151633966044770966962, 8.23162174165622957018334950182, 8.54511628922507368892737112644, 9.52169858083593216929619098894, 10.4173213782681099740299832451, 10.643334129618819153200824704589, 12.09070145065442245312007723168, 12.68732423766149425195637786549, 13.6526110958587792365285894727, 14.01711444756874974630637844561, 14.80527701178853753155138230008, 15.41416793764548813663136196293, 16.21641125809244259719139606943, 17.08183007591109674037835769695, 18.20880347192089431571433934265, 18.45269281264878459496770647606, 18.93161752735886565810239475270, 20.34046434223843793760187501757

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