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Properties

Label 1-407-407.183-r0-0-0
Degree $1$
Conductor $407$
Sign $0.662 + 0.749i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (0.139 + 0.990i)2-s + (−0.438 + 0.898i)3-s + (−0.961 + 0.275i)4-s + (0.927 − 0.374i)5-s + (−0.951 − 0.309i)6-s + (0.241 − 0.970i)7-s + (−0.406 − 0.913i)8-s + (−0.615 − 0.788i)9-s + (0.5 + 0.866i)10-s + (0.173 − 0.984i)12-s + (−0.999 − 0.0348i)13-s + (0.994 + 0.104i)14-s + (−0.0697 + 0.997i)15-s + (0.848 − 0.529i)16-s + (0.999 − 0.0348i)17-s + (0.694 − 0.719i)18-s + ⋯
L(s)  = 1  + (0.139 + 0.990i)2-s + (−0.438 + 0.898i)3-s + (−0.961 + 0.275i)4-s + (0.927 − 0.374i)5-s + (−0.951 − 0.309i)6-s + (0.241 − 0.970i)7-s + (−0.406 − 0.913i)8-s + (−0.615 − 0.788i)9-s + (0.5 + 0.866i)10-s + (0.173 − 0.984i)12-s + (−0.999 − 0.0348i)13-s + (0.994 + 0.104i)14-s + (−0.0697 + 0.997i)15-s + (0.848 − 0.529i)16-s + (0.999 − 0.0348i)17-s + (0.694 − 0.719i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.095683585 + 0.4940990311i\)
\(L(\frac12)\) \(\approx\) \(1.095683585 + 0.4940990311i\)
\(L(1)\) \(\approx\) \(0.9169763344 + 0.4721143568i\)
\(L(1)\) \(\approx\) \(0.9169763344 + 0.4721143568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.139 + 0.990i)T \)
3 \( 1 + (-0.438 + 0.898i)T \)
5 \( 1 + (0.927 - 0.374i)T \)
7 \( 1 + (0.241 - 0.970i)T \)
13 \( 1 + (-0.999 - 0.0348i)T \)
17 \( 1 + (0.999 - 0.0348i)T \)
19 \( 1 + (-0.898 - 0.438i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.994 - 0.104i)T \)
31 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.559 + 0.829i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (-0.374 + 0.927i)T \)
59 \( 1 + (0.0697 - 0.997i)T \)
61 \( 1 + (-0.788 - 0.615i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.990 + 0.139i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.529 - 0.848i)T \)
83 \( 1 + (-0.0348 - 0.999i)T \)
89 \( 1 + (0.642 - 0.766i)T \)
97 \( 1 + (0.743 + 0.669i)T \)
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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

−24.14978829925497381072009912047, −23.056248690805626710777266324976, −22.46949950229105825468605089448, −21.40371973489988653077276586480, −21.16362161530654198152326791976, −19.51389837867402739482276977020, −19.07092127473403868087617282676, −18.19967917496982804307564926179, −17.56908700949701435319738182783, −16.79852338791486435183641847897, −14.93720341676613095570885570942, −14.27858526418825469482040435908, −13.35178351609317185300307022689, −12.423396640054653655002267209000, −11.953274340972418605330820527502, −10.83573416094449018586897615474, −9.99648446080061210077935738573, −8.96572802044917614350677971027, −7.92302025096897875166624240427, −6.526098297331043845803962839835, −5.58589508984071312246465192846, −4.89674645391363098736983231578, −2.96544442004104782706363659188, −2.26523920623497224112237915829, −1.300276545883718988474420674067, 0.818037033779406961847772200935, 2.98336013623069331399538212033, 4.486343315517340652845321816285, 4.83153951741292258190547754057, 5.96603464069462442087436567943, 6.81397850798190852852264596822, 8.04652385037245009928744938102, 9.1409488694752221606286180690, 9.94026782420522577787562139117, 10.61642505486509437278075256180, 12.15039742045935637189501460486, 13.08838378029705440979553377281, 14.184138291565212451081696420163, 14.66023105784216009919338637520, 15.77486941328388244525478095235, 16.80724694307183851096609055430, 17.123338261168675133084590672331, 17.72187106852284616331855967047, 19.10140514067862834674124199235, 20.431994398367981623275714537370, 21.26899761564518496974183962404, 21.8420969808301538435451284620, 22.88187111842830501994011211975, 23.484480831982575532683379757108, 24.44810956467106150625445837398

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