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Properties

Label 1-407-407.172-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.539 + 0.841i$
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  + (−0.999 − 0.0348i)2-s + (−0.961 − 0.275i)3-s + (0.997 + 0.0697i)4-s + (−0.469 + 0.882i)5-s + (0.951 + 0.309i)6-s + (−0.438 − 0.898i)7-s + (−0.994 − 0.104i)8-s + (0.848 + 0.529i)9-s + (0.5 − 0.866i)10-s + (−0.939 − 0.342i)12-s + (−0.927 − 0.374i)13-s + (0.406 + 0.913i)14-s + (0.694 − 0.719i)15-s + (0.990 + 0.139i)16-s + (0.927 − 0.374i)17-s + (−0.829 − 0.559i)18-s + ⋯
L(s)  = 1  + (−0.999 − 0.0348i)2-s + (−0.961 − 0.275i)3-s + (0.997 + 0.0697i)4-s + (−0.469 + 0.882i)5-s + (0.951 + 0.309i)6-s + (−0.438 − 0.898i)7-s + (−0.994 − 0.104i)8-s + (0.848 + 0.529i)9-s + (0.5 − 0.866i)10-s + (−0.939 − 0.342i)12-s + (−0.927 − 0.374i)13-s + (0.406 + 0.913i)14-s + (0.694 − 0.719i)15-s + (0.990 + 0.139i)16-s + (0.927 − 0.374i)17-s + (−0.829 − 0.559i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.539 + 0.841i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.539 + 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09081590845 + 0.1660729148i\)
\(L(\frac12)\) \(\approx\) \(0.09081590845 + 0.1660729148i\)
\(L(1)\) \(\approx\) \(0.3901676211 + 0.01953401047i\)
\(L(1)\) \(\approx\) \(0.3901676211 + 0.01953401047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.999 - 0.0348i)T \)
3 \( 1 + (-0.961 - 0.275i)T \)
5 \( 1 + (-0.469 + 0.882i)T \)
7 \( 1 + (-0.438 - 0.898i)T \)
13 \( 1 + (-0.927 - 0.374i)T \)
17 \( 1 + (0.927 - 0.374i)T \)
19 \( 1 + (-0.275 + 0.961i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.406 - 0.913i)T \)
31 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (-0.882 + 0.469i)T \)
59 \( 1 + (-0.694 + 0.719i)T \)
61 \( 1 + (-0.529 - 0.848i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.139 + 0.990i)T \)
83 \( 1 + (0.374 + 0.927i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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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

−24.04394846813565569022218406652, −23.46616313428158708978457039178, −22.08682336105017077360047508739, −21.44483981639303206357167774952, −20.49931295623942367437899811917, −19.43528389174537950927540855794, −18.81920714479510855923151582033, −17.814994435364061235888857166409, −16.83155847872010031734536248471, −16.494545135873373559763589960468, −15.51588430802525414305523088868, −14.87321206260203747824600363229, −12.74671923530203956974986473819, −12.28113155198610156657573631477, −11.48568322277291856722339976493, −10.49809518976379652482742084364, −9.396002024935715942047577876649, −8.91225431121645771799141151630, −7.61453658900787539340092502893, −6.645549207249063502581646820165, −5.57903505706978105670289898205, −4.72242759556015622759169540165, −3.13255634230723438166046228723, −1.6060012484334309644902363984, −0.198290756706366598934982713370, 1.17537029357745699329475440960, 2.71648652143706731691889777538, 3.892062690508015446092621384659, 5.52134830771915625938927523494, 6.58812739281655452540684711491, 7.35724806162544518860492828099, 7.84052984558542085862758009155, 9.656366689338122358774750308485, 10.31827408870801958832219265898, 10.97264918582793321382785694535, 11.92965564199151411000072144536, 12.67369460255013314165146331361, 14.1403458630164957194639293038, 15.22125125056257747236892527338, 16.15924568513632576796576518259, 16.9467475986343971282408171687, 17.54382297135798340390473348602, 18.63241205893375247515527389509, 19.10646515355346261146105326815, 19.95788295632572262331528024030, 21.10262745357879623097520009539, 22.1436743914515771259224614738, 23.102358900913080321704897207914, 23.54566205520073363803874438142, 24.75729771989808891369137884867

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