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Properties

Label 1-403-403.6-r0-0-0
Degree $1$
Conductor $403$
Sign $0.978 + 0.204i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (−0.866 + 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + 9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + 9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.978 + 0.204i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.978 + 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5523654168 + 0.05718897400i\)
\(L(\frac12)\) \(\approx\) \(0.5523654168 + 0.05718897400i\)
\(L(1)\) \(\approx\) \(0.5315564690 + 0.08974834694i\)
\(L(1)\) \(\approx\) \(0.5315564690 + 0.08974834694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−24.28063467051295976544510776539, −23.66190575237257211938297605125, −22.384418442893795949027768820033, −21.58726587553418703506813539536, −20.96959164796431432584015384989, −19.658046641864692197947157095, −19.21718313514803346785585109922, −18.10895026483732315347476700093, −17.34097666399084174512088029967, −16.75298836831092399579039064869, −15.74594479548110680108441418154, −15.045867834677908437131725905009, −13.24587408043918044736704632176, −12.29418453154215495817900128492, −11.50827920081843144160623824025, −11.22694750063332064432111122673, −10.01989661812643834518249792100, −8.77193987916013797107753700928, −8.18517479022642103664332545181, −7.03111285904467525589675566637, −5.97284788024202056613260438928, −4.56855339755178631579363345474, −3.79293349135182489030410170762, −1.97689412748316285901164136623, −0.89424416888779005724894885547, 0.72357518784284036816999257817, 2.049119844448536838006919582168, 4.10909926939803474051404330593, 4.89790767833420165333974377770, 6.30703200601255302354698908356, 7.00906517069983881665740819573, 7.75736096676191916081028297975, 8.868921318432395220100688510630, 10.177112188894180677613162325405, 10.8833350000466220777574669528, 11.55163645374821864050042377026, 12.40344701270989866063896638191, 14.20479757945119632819775976668, 14.820518913094941255390148935493, 15.869827816847359666957033847036, 16.553750614081088455004701710032, 17.47853468462026372113404418573, 18.07135567386143042097928379895, 18.90063997716903925712764946600, 19.88356624537387601328476707817, 20.67783540491763681093844585641, 22.036083020698393318066113952766, 23.00330045782449400944786683727, 23.497888828468105478251245221596, 24.361115490377515329521566224378

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