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Properties

Label 1-399-399.395-r1-0-0
Degree $1$
Conductor $399$
Sign $0.937 + 0.347i$
Analytic cond. $42.8785$
Root an. cond. $42.8785$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)10-s − 11-s + (−0.939 − 0.342i)13-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + 20-s + (0.939 − 0.342i)22-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + 26-s + (0.766 − 0.642i)29-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)10-s − 11-s + (−0.939 − 0.342i)13-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + 20-s + (0.939 − 0.342i)22-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + 26-s + (0.766 − 0.642i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(42.8785\)
Root analytic conductor: \(42.8785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 399,\ (1:\ ),\ 0.937 + 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254000612 + 0.2252174286i\)
\(L(\frac12)\) \(\approx\) \(1.254000612 + 0.2252174286i\)
\(L(1)\) \(\approx\) \(0.7855287969 + 0.1369561799i\)
\(L(1)\) \(\approx\) \(0.7855287969 + 0.1369561799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−24.38650954814427287506286957514, −23.47030002064475387771635825436, −21.90492249938035638688651668563, −21.41368586347260322759163634410, −20.58226865156193004991800700378, −19.78058195635008967168436920301, −18.8224015711691850321225276015, −18.00436075823890785145891810804, −17.09382565358337792253880197787, −16.60838642465971397968492142304, −15.52774661826906139339059889821, −14.45334952106697164674983640882, −12.99192134227967207601550486538, −12.65155407375311254212007386090, −11.39896938615838689833966543259, −10.34578342944866743625747266341, −9.71998197856093508135928946488, −8.73869176040471168117782576339, −7.91549785279707908686272594464, −6.8007471700253725560919605590, −5.66076077994227819655291402265, −4.49170822860422932261194977768, −2.8753347824751022056112940959, −1.98037768929550917935183286323, −0.76120610990519622452618479344, 0.68719491040264996025327985609, 2.26032129271243150525307382661, 2.910572983888951383825609112170, 5.06466998460262951707141611191, 5.7894946805833606370445673788, 7.06685547338718966504072562690, 7.55704810733953281945560967837, 8.89269215143453886398888261866, 9.78681342995556388034367587247, 10.4593678270389922698472740306, 11.31305222801200757078517576037, 12.57851028886919434696179589393, 13.79723383118444608757441857275, 14.6397413815945488064039048710, 15.50493204575097727231165396850, 16.39991449904610682920671943969, 17.491683613707708671744611133105, 17.947039956140438602062141388653, 18.83343473173550475124236056764, 19.61979613468664264376558753681, 20.764285660395758836962147986891, 21.38866211332100300770691538658, 22.613874483864489623191274960692, 23.43369123203846651167518395230, 24.55611246776974582769096765125

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