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Properties

Label 1-311-311.41-r1-0-0
Degree $1$
Conductor $311$
Sign $-0.887 + 0.460i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
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  + (−0.758 + 0.651i)2-s + (−0.758 − 0.651i)3-s + (0.151 − 0.988i)4-s + (0.820 − 0.571i)5-s + 6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (0.151 + 0.988i)9-s + (−0.250 + 0.968i)10-s + (−0.347 − 0.937i)11-s + (−0.758 + 0.651i)12-s + (0.151 + 0.988i)13-s + (0.979 + 0.201i)14-s + (−0.994 − 0.101i)15-s + (−0.954 − 0.299i)16-s + (0.250 − 0.968i)17-s + ⋯
L(s)  = 1  + (−0.758 + 0.651i)2-s + (−0.758 − 0.651i)3-s + (0.151 − 0.988i)4-s + (0.820 − 0.571i)5-s + 6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (0.151 + 0.988i)9-s + (−0.250 + 0.968i)10-s + (−0.347 − 0.937i)11-s + (−0.758 + 0.651i)12-s + (0.151 + 0.988i)13-s + (0.979 + 0.201i)14-s + (−0.994 − 0.101i)15-s + (−0.954 − 0.299i)16-s + (0.250 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ -0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06616803983 - 0.2712253485i\)
\(L(\frac12)\) \(\approx\) \(-0.06616803983 - 0.2712253485i\)
\(L(1)\) \(\approx\) \(0.5103146662 - 0.1542597265i\)
\(L(1)\) \(\approx\) \(0.5103146662 - 0.1542597265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.758 + 0.651i)T \)
3 \( 1 + (-0.758 - 0.651i)T \)
5 \( 1 + (0.820 - 0.571i)T \)
7 \( 1 + (-0.612 - 0.790i)T \)
11 \( 1 + (-0.347 - 0.937i)T \)
13 \( 1 + (0.151 + 0.988i)T \)
17 \( 1 + (0.250 - 0.968i)T \)
19 \( 1 + (-0.820 - 0.571i)T \)
23 \( 1 + (-0.528 - 0.848i)T \)
29 \( 1 + (-0.979 + 0.201i)T \)
31 \( 1 + (0.250 + 0.968i)T \)
37 \( 1 + (0.612 - 0.790i)T \)
41 \( 1 + (-0.688 - 0.724i)T \)
43 \( 1 + (0.612 + 0.790i)T \)
47 \( 1 + (0.979 - 0.201i)T \)
53 \( 1 + (-0.612 - 0.790i)T \)
59 \( 1 + (0.612 + 0.790i)T \)
61 \( 1 + (-0.820 - 0.571i)T \)
67 \( 1 + (0.688 - 0.724i)T \)
71 \( 1 + (-0.918 - 0.394i)T \)
73 \( 1 + (-0.440 + 0.897i)T \)
79 \( 1 + (0.688 + 0.724i)T \)
83 \( 1 + (-0.994 + 0.101i)T \)
89 \( 1 + (-0.612 + 0.790i)T \)
97 \( 1 + (-0.918 - 0.394i)T \)
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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

−25.69170645048087334849334190404, −25.187998659060017083245914055425, −23.45116478249365985944410823001, −22.34685706330433859124735581041, −22.03759975224951231576936271018, −21.069782569634151381291159657, −20.34776652509334223154707825761, −18.97670117445343620476837573583, −18.285299895198248381138447088140, −17.44028768262753156174366160668, −16.863997521108950391250688938985, −15.535049547987703915384462895530, −15.00097027190658107988591219735, −13.11630228407000628737828113026, −12.515955857145587510269112140269, −11.44603688805371182413608337753, −10.30100351861444245770219839392, −10.02275961077327894548776389491, −9.07102312264631443545942839876, −7.70857205212652500110376522360, −6.33454591928848083857715013776, −5.61078583857283897902765648384, −4.00430738259928686312646491325, −2.87815572986456323067157019702, −1.71451534227383525120891463327, 0.135510942139527303091972956651, 1.0293283846891661048706384008, 2.29887035110136441795346496033, 4.55990623145062181215695519693, 5.64608641498914906510328155423, 6.45590927692030964555463265947, 7.19654700037837547021357595986, 8.452130494306595763547343630700, 9.41027390247972717225290447627, 10.46260215764079425311549605128, 11.219905930406515368894994763786, 12.62419327700578431069649905467, 13.65374596176268540296428726316, 14.12901297533178452826497803598, 16.04988534739322394061255100592, 16.48944047688587874492912552460, 17.1138637095415059349898897690, 18.08911431681421859047055068954, 18.839778818181072093202594954936, 19.687067363603625347465358166499, 20.816418043537974497779538410813, 21.98097417356215825370763014986, 23.074936092586366614985449055053, 23.894654062340149611560639627290, 24.397928963376631059165232512143

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