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Properties

Label 1-3407-3407.1002-r0-0-0
Degree $1$
Conductor $3407$
Sign $-0.991 + 0.128i$
Analytic cond. $15.8220$
Root an. cond. $15.8220$
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  + (−0.954 + 0.299i)2-s + (0.397 + 0.917i)3-s + (0.820 − 0.571i)4-s + (0.0985 + 0.995i)5-s + (−0.654 − 0.756i)6-s + (0.627 − 0.778i)7-s + (−0.611 + 0.791i)8-s + (−0.684 + 0.729i)9-s + (−0.392 − 0.919i)10-s + (0.186 + 0.982i)11-s + (0.850 + 0.525i)12-s + (−0.256 − 0.966i)13-s + (−0.364 + 0.931i)14-s + (−0.874 + 0.485i)15-s + (0.345 − 0.938i)16-s + (0.976 + 0.215i)17-s + ⋯
L(s)  = 1  + (−0.954 + 0.299i)2-s + (0.397 + 0.917i)3-s + (0.820 − 0.571i)4-s + (0.0985 + 0.995i)5-s + (−0.654 − 0.756i)6-s + (0.627 − 0.778i)7-s + (−0.611 + 0.791i)8-s + (−0.684 + 0.729i)9-s + (−0.392 − 0.919i)10-s + (0.186 + 0.982i)11-s + (0.850 + 0.525i)12-s + (−0.256 − 0.966i)13-s + (−0.364 + 0.931i)14-s + (−0.874 + 0.485i)15-s + (0.345 − 0.938i)16-s + (0.976 + 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3407\)
Sign: $-0.991 + 0.128i$
Analytic conductor: \(15.8220\)
Root analytic conductor: \(15.8220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3407} (1002, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3407,\ (0:\ ),\ -0.991 + 0.128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08308030937 + 1.287944882i\)
\(L(\frac12)\) \(\approx\) \(0.08308030937 + 1.287944882i\)
\(L(1)\) \(\approx\) \(0.6718224382 + 0.5773164812i\)
\(L(1)\) \(\approx\) \(0.6718224382 + 0.5773164812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3407 \( 1 \)
good2 \( 1 + (-0.954 + 0.299i)T \)
3 \( 1 + (0.397 + 0.917i)T \)
5 \( 1 + (0.0985 + 0.995i)T \)
7 \( 1 + (0.627 - 0.778i)T \)
11 \( 1 + (0.186 + 0.982i)T \)
13 \( 1 + (-0.256 - 0.966i)T \)
17 \( 1 + (0.976 + 0.215i)T \)
19 \( 1 + (0.300 + 0.953i)T \)
23 \( 1 + (-0.0599 + 0.998i)T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (-0.180 + 0.983i)T \)
37 \( 1 + (0.990 + 0.139i)T \)
41 \( 1 + (0.887 + 0.461i)T \)
43 \( 1 + (-0.659 + 0.751i)T \)
47 \( 1 + (-0.976 - 0.217i)T \)
53 \( 1 + (-0.249 - 0.968i)T \)
59 \( 1 + (-0.0819 - 0.996i)T \)
61 \( 1 + (0.549 + 0.835i)T \)
67 \( 1 + (-0.977 + 0.210i)T \)
71 \( 1 + (0.998 + 0.0553i)T \)
73 \( 1 + (0.999 + 0.0295i)T \)
79 \( 1 + (-0.432 + 0.901i)T \)
83 \( 1 + (0.463 - 0.885i)T \)
89 \( 1 + (0.400 - 0.916i)T \)
97 \( 1 + (0.999 - 0.0258i)T \)
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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

−18.58323832104626829182347815943, −17.957233500111988649845429755797, −17.09641058009600159216663855700, −16.71640556997975386635731258201, −15.93851886031701720236767183955, −15.05711828140379050706167619977, −14.24282631883419753669103645693, −13.501258734992206328895883557895, −12.734216673163906966256736478064, −11.962803896827211025676233457994, −11.702155696986147390395995387713, −10.95681487002539291889940886232, −9.552361095478831651548288282534, −9.24156467155500052501239822409, −8.5374124706723852458082334035, −8.02919443730342114118846184798, −7.39914377295750582599069873005, −6.305721020642219794660639449151, −5.83908671140980498600524578690, −4.7228674056127303812551910207, −3.66430126595907509632080247630, −2.57313283410318445096539665046, −2.131411796909127770124644807327, −1.16557341866694492475356175622, −0.53541271195130202217217948241, 1.27162079032931972167180702358, 2.03747277200072463510920398897, 3.1419199297531813658704104132, 3.576745111581871779043402704637, 4.83781196028111412234942754310, 5.46801965843903057741306794911, 6.39881563047447762487480798417, 7.43939462046459304132107183522, 7.70734134777389015465862582116, 8.400633339357793524317081815812, 9.61311075370124366288978146801, 9.97509077548108019300701709539, 10.407644691124406447562031794667, 11.1460986696598873864624183590, 11.756623210793127667644219889756, 12.908051474035921360822052228589, 14.16036981906503729906752122696, 14.56258088064935242673102450920, 14.87891881623631579578087157398, 15.71090815752548074013724838733, 16.436304277320594280204274854108, 17.088577871398656018644985237494, 17.88039775951059954997711579343, 18.1137730409452605196780206995, 19.24241885780189911595423831643

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