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Properties

Label 1-4031-4031.4-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.953 - 0.299i$
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  + (0.857 + 0.514i)2-s + (0.953 − 0.300i)3-s + (0.471 + 0.881i)4-s + (0.413 − 0.910i)5-s + (0.972 + 0.232i)6-s + (−0.870 − 0.491i)7-s + (−0.0487 + 0.998i)8-s + (0.818 − 0.574i)9-s + (0.822 − 0.568i)10-s + (0.759 − 0.651i)11-s + (0.715 + 0.699i)12-s + (0.228 + 0.973i)13-s + (−0.494 − 0.869i)14-s + (0.120 − 0.992i)15-s + (−0.555 + 0.831i)16-s + (0.949 + 0.313i)17-s + ⋯
L(s)  = 1  + (0.857 + 0.514i)2-s + (0.953 − 0.300i)3-s + (0.471 + 0.881i)4-s + (0.413 − 0.910i)5-s + (0.972 + 0.232i)6-s + (−0.870 − 0.491i)7-s + (−0.0487 + 0.998i)8-s + (0.818 − 0.574i)9-s + (0.822 − 0.568i)10-s + (0.759 − 0.651i)11-s + (0.715 + 0.699i)12-s + (0.228 + 0.973i)13-s + (−0.494 − 0.869i)14-s + (0.120 − 0.992i)15-s + (−0.555 + 0.831i)16-s + (0.949 + 0.313i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.953 - 0.299i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.953 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.759005385 - 0.7303600367i\)
\(L(\frac12)\) \(\approx\) \(4.759005385 - 0.7303600367i\)
\(L(1)\) \(\approx\) \(2.518733429 + 0.02190421570i\)
\(L(1)\) \(\approx\) \(2.518733429 + 0.02190421570i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.857 + 0.514i)T \)
3 \( 1 + (0.953 - 0.300i)T \)
5 \( 1 + (0.413 - 0.910i)T \)
7 \( 1 + (-0.870 - 0.491i)T \)
11 \( 1 + (0.759 - 0.651i)T \)
13 \( 1 + (0.228 + 0.973i)T \)
17 \( 1 + (0.949 + 0.313i)T \)
19 \( 1 + (-0.985 - 0.168i)T \)
23 \( 1 + (0.425 - 0.905i)T \)
31 \( 1 + (0.741 - 0.670i)T \)
37 \( 1 + (-0.701 + 0.712i)T \)
41 \( 1 + (0.974 - 0.225i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (0.272 + 0.962i)T \)
53 \( 1 + (-0.0162 + 0.999i)T \)
59 \( 1 + (-0.334 + 0.942i)T \)
61 \( 1 + (0.783 - 0.620i)T \)
67 \( 1 + (0.943 - 0.331i)T \)
71 \( 1 + (0.663 - 0.748i)T \)
73 \( 1 + (-0.643 - 0.765i)T \)
79 \( 1 + (0.864 + 0.502i)T \)
83 \( 1 + (0.847 - 0.530i)T \)
89 \( 1 + (-0.880 - 0.474i)T \)
97 \( 1 + (0.988 - 0.149i)T \)
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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

−18.82494331976935208758893981519, −18.085592191083483353310190496993, −17.150122286897484792391216013340, −16.01472386689459021684368160388, −15.56802857743659767277603901989, −14.75325684216886301899853444322, −14.609064649175404254934467609862, −13.73269569646562877528119977610, −13.08269089993499255501801726222, −12.55996256512574301059508954893, −11.75109773469621551252835869491, −10.811711326518431344260556334484, −10.11278194964589365293903406961, −9.77089971128414872569362632226, −9.07706256325957143356936739806, −8.01473374581704005170973271951, −7.014935894140638135412298833964, −6.59872099378333532415849049633, −5.66301381053114561018428254487, −5.01744956517357072175036489943, −3.70950748840629742835539088762, −3.56377568874545619658981391015, −2.69844981592032034902172625172, −2.16780552327394595377411504216, −1.210633522091489057364073177632, 0.89094858898899564839925576208, 1.81756488447403871296837279170, 2.6974558379174273476243092501, 3.54821813865925599110688964393, 4.13605807696987132159443218508, 4.720396213134305003349175039025, 6.03616031697874673238910516800, 6.3526552106971891215609991260, 7.0719653252720838073358185385, 7.99629229788212289767190318797, 8.663244065289313749407560033828, 9.13743166999395811340936213438, 9.95613349022681146607390779175, 10.93958028499660432945054527824, 12.09424304169944113704232887832, 12.44735113653013517973953919253, 13.15586719338403114772928550797, 13.78285351152389468657059668152, 14.08705535592207198301137191128, 14.89950661640843924289416402140, 15.67857695388307511098575709083, 16.4465877442093778556095338641, 16.84094308497511197497872828391, 17.36584200761539619606978963691, 18.69061086321579798012897730781

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