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Properties

Label 1-4031-4031.1433-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.184 - 0.982i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.5 − 0.866i)9-s + i·10-s + (−0.866 − 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s − i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.5 − 0.866i)9-s + i·10-s + (−0.866 − 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s − i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (1433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.184 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1537139099 + 0.1851896651i\)
\(L(\frac12)\) \(\approx\) \(-0.1537139099 + 0.1851896651i\)
\(L(1)\) \(\approx\) \(0.9000072394 + 0.5792084782i\)
\(L(1)\) \(\approx\) \(0.9000072394 + 0.5792084782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 - T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−18.01393149864829202977213875875, −17.49251792910444007646687269143, −16.31943757637712723537473851057, −15.83091574577584854418697044514, −15.64101033281183536591561737748, −14.24622125665011776658788384040, −13.52783168128022984606187417684, −13.204930660094881369286079557095, −12.44024025662245275984655391175, −11.931775574189782053329146804079, −11.46840624642580210859783994938, −10.47063896061955124668198049733, −9.849296361908112787357597653722, −9.13926932015478493045948278956, −8.25586322932100653755747610848, −7.025380405547582551002966881842, −6.5695228763685084568684589105, −5.72461574844881877362922753182, −5.210480243580613368758021817740, −4.71865768014499375180106278381, −3.77298545698512217533860863807, −2.44325832667568274868802612005, −2.11742738167707171784167738984, −1.205411761797555015086026271658, −0.05270331630182916683369336072, 1.45115202362739966251651802041, 2.7649390085889292054553530891, 3.355183343164111310862807860905, 4.00722172519574823889985469672, 4.87730595350859028069792230258, 5.731588636827265558922602145074, 6.15406005039352618188485658981, 6.71737592077124699886369094213, 7.59491858081692572377135470323, 8.21745180326764623935359989290, 9.51056028458096025277055426531, 10.34823347049600673427084998705, 10.65132481915591131638586050239, 11.3490699040118972237266687763, 12.14910164881937067554309843278, 13.06592851691232758879335377901, 13.48653002604702312404070645, 14.1000347812350437785383458229, 15.02294913753438711628711013484, 15.68317492953542238814093238758, 16.01322262984685001471618620363, 16.847596024703302729341437006663, 17.49459659652197561413951543007, 17.99122181530503836363774868580, 18.69337482896166022447751240021

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