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Properties

Label 1-4031-4031.3057-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.981 + 0.189i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
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  − i·2-s − i·3-s − 4-s − 5-s − 6-s + 7-s + i·8-s − 9-s + i·10-s + i·11-s + i·12-s − 13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 5-s − 6-s + 7-s + i·8-s − 9-s + i·10-s + i·11-s + i·12-s − 13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.981 + 0.189i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (3057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.981 + 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1765921698 + 0.01684946637i\)
\(L(\frac12)\) \(\approx\) \(0.1765921698 + 0.01684946637i\)
\(L(1)\) \(\approx\) \(0.4731946696 - 0.4919024128i\)
\(L(1)\) \(\approx\) \(0.4731946696 - 0.4919024128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + iT \)
31 \( 1 + iT \)
37 \( 1 + iT \)
41 \( 1 + iT \)
43 \( 1 - T \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + iT \)
71 \( 1 - iT \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 + T \)
89 \( 1 - T \)
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

−18.31719473103228153915336688630, −17.51875594844318908856727565152, −16.76863934141296262108714911444, −16.485086046837179731808711831625, −15.70751510649586921795597156216, −15.006547087356092473668543057556, −14.63393057465700070715013398, −14.14005493363210811110166513471, −13.15681896755518717422593658886, −12.11267925140306792644758626019, −11.625144525345132130008418835769, −10.73955403955334654067938200724, −10.14270582709123471046686556122, −9.2674132674277495115298465276, −8.48216547846303534405804439447, −7.905951316468217880128483385471, −7.68692910206936376133861516481, −6.188774808221313469984199569556, −5.86486075157497076293230400135, −4.69680211052594990618837202293, −4.51069279200372492761634075122, −3.67769459870229515185230973792, −2.94660733366477346033144175063, −1.42665548747340695863350493449, −0.06721657915031830436442843613, 0.800689865310719206536340167903, 1.886561242989667581989525021209, 2.34752075963086257188655583154, 3.21751045389474754677912704993, 4.24771816172325036720010348755, 4.88699543285372651087649189688, 5.42452978696337500616170795258, 6.9430044302705494291304598768, 7.37869333106149339676605468053, 7.99902318242035922733746374582, 8.76114442647719600940188451217, 9.41447804465079867107141064000, 10.482995167674452413679473956659, 11.12188979101641546340195449381, 11.84153048399223414790159473492, 12.18925460159952063487426362239, 12.64897225067042564271385425902, 13.7315334455166264619849496233, 14.19558176725506374551044991212, 14.85885829549634588275569529397, 15.60202434897469201939562491473, 16.74741177135596385007650730573, 17.638108176589933847357513965003, 17.80165174689606531232162419124, 18.57273928483982788607309435846

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