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Properties

Label 1-3997-3997.237-r1-0-0
Degree $1$
Conductor $3997$
Sign $0.457 - 0.889i$
Analytic cond. $429.537$
Root an. cond. $429.537$
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  + (−0.821 − 0.569i)2-s + (−0.938 − 0.345i)3-s + (0.350 + 0.936i)4-s + (0.360 + 0.932i)5-s + (0.574 + 0.818i)6-s + (0.245 − 0.969i)8-s + (0.761 + 0.648i)9-s + (0.234 − 0.972i)10-s + (0.959 + 0.282i)11-s + (−0.00551 − 0.999i)12-s + (−0.0715 + 0.997i)13-s + (−0.0165 − 0.999i)15-s + (−0.754 + 0.656i)16-s + (−0.391 − 0.920i)17-s + (−0.256 − 0.966i)18-s + (0.556 + 0.831i)19-s + ⋯
L(s)  = 1  + (−0.821 − 0.569i)2-s + (−0.938 − 0.345i)3-s + (0.350 + 0.936i)4-s + (0.360 + 0.932i)5-s + (0.574 + 0.818i)6-s + (0.245 − 0.969i)8-s + (0.761 + 0.648i)9-s + (0.234 − 0.972i)10-s + (0.959 + 0.282i)11-s + (−0.00551 − 0.999i)12-s + (−0.0715 + 0.997i)13-s + (−0.0165 − 0.999i)15-s + (−0.754 + 0.656i)16-s + (−0.391 − 0.920i)17-s + (−0.256 − 0.966i)18-s + (0.556 + 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $0.457 - 0.889i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ 0.457 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7715134394 - 0.4705033183i\)
\(L(\frac12)\) \(\approx\) \(0.7715134394 - 0.4705033183i\)
\(L(1)\) \(\approx\) \(0.6135739523 - 0.05761313731i\)
\(L(1)\) \(\approx\) \(0.6135739523 - 0.05761313731i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.821 - 0.569i)T \)
3 \( 1 + (-0.938 - 0.345i)T \)
5 \( 1 + (0.360 + 0.932i)T \)
11 \( 1 + (0.959 + 0.282i)T \)
13 \( 1 + (-0.0715 + 0.997i)T \)
17 \( 1 + (-0.391 - 0.920i)T \)
19 \( 1 + (0.556 + 0.831i)T \)
23 \( 1 + (0.371 - 0.928i)T \)
29 \( 1 + (0.137 + 0.990i)T \)
31 \( 1 + (-0.945 - 0.324i)T \)
37 \( 1 + (-0.644 - 0.764i)T \)
41 \( 1 + (0.998 - 0.0550i)T \)
43 \( 1 + (-0.997 + 0.0770i)T \)
47 \( 1 + (0.191 + 0.981i)T \)
53 \( 1 + (0.287 + 0.957i)T \)
59 \( 1 + (0.677 - 0.735i)T \)
61 \( 1 + (0.982 + 0.186i)T \)
67 \( 1 + (0.685 - 0.728i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.984 + 0.175i)T \)
79 \( 1 + (-0.0605 - 0.998i)T \)
83 \( 1 + (0.421 - 0.906i)T \)
89 \( 1 + (-0.996 + 0.0880i)T \)
97 \( 1 + (0.782 - 0.622i)T \)
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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

−17.935279775500569763576274962194, −17.66061295836011664839467953686, −17.1010236925884841447846421568, −16.59664331232190610585147756474, −15.866872588612262846511468758718, −15.32118460154293198117660911196, −14.68289112544601664016927803961, −13.53722693624063046212406333014, −13.03685971474725912164403454366, −12.00592579323681787949518071612, −11.48664912005200560444844005698, −10.73607949161671686605386378420, −9.96554277616946071021319810493, −9.462768005315687931588957120059, −8.75579772021984607825794801146, −8.10837648087182897785573008799, −7.06830579857831803117661453291, −6.50053131771622694168531625012, −5.51369649192720575308659578130, −5.427917237587704773102521261226, −4.42331687341416222773374344071, −3.52889761017018552425394603255, −2.07529696874488496846245244798, −1.1627272085825319756544762712, −0.68898155884270382642918496250, 0.30073939432251059717543231934, 1.3907148629811153055924953905, 1.92255345984161103134707843276, 2.79722657918345900407945513255, 3.79863411270022386120738398349, 4.52109816711243289544586396448, 5.65006333833684571867510754229, 6.502923866029961666564771866884, 7.0752959132060101706815119580, 7.362374599796194595702755785070, 8.59814231873089265799917732913, 9.44674113721715019417457116333, 9.861059088902146964934575505288, 10.85072403149959839748600216467, 11.12576989369409464436554909894, 11.89788258031857766928984330315, 12.37081491535459929708904038671, 13.22712337835356659720436036765, 14.12505918338212762144424963622, 14.63775413467393439128865727997, 15.9123933833656356599865996073, 16.35902495419399789845364829773, 17.01563689439541664743223404476, 17.704739844212593814124531663304, 18.17769499758166641025694873294

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