↑

Properties

Label 1-367-367.28-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.331 - 0.943i$
Analytic cond. $1.70434$
Root an. cond. $1.70434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.999 + 0.0171i)2-s + (0.679 + 0.733i)3-s + (0.999 − 0.0343i)4-s + (0.229 − 0.973i)5-s + (−0.691 − 0.722i)6-s + (−0.967 − 0.254i)7-s + (−0.998 + 0.0514i)8-s + (−0.0771 + 0.997i)9-s + (−0.212 + 0.977i)10-s + (−0.909 + 0.416i)11-s + (0.704 + 0.710i)12-s + (−0.824 − 0.565i)13-s + (0.971 + 0.238i)14-s + (0.870 − 0.492i)15-s + (0.997 − 0.0686i)16-s + (−0.762 − 0.647i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0171i)2-s + (0.679 + 0.733i)3-s + (0.999 − 0.0343i)4-s + (0.229 − 0.973i)5-s + (−0.691 − 0.722i)6-s + (−0.967 − 0.254i)7-s + (−0.998 + 0.0514i)8-s + (−0.0771 + 0.997i)9-s + (−0.212 + 0.977i)10-s + (−0.909 + 0.416i)11-s + (0.704 + 0.710i)12-s + (−0.824 − 0.565i)13-s + (0.971 + 0.238i)14-s + (0.870 − 0.492i)15-s + (0.997 − 0.0686i)16-s + (−0.762 − 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2699219393 - 0.3810262663i\)
\(L(\frac12)\) \(\approx\) \(0.2699219393 - 0.3810262663i\)
\(L(1)\) \(\approx\) \(0.6239871620 - 0.06919865659i\)
\(L(1)\) \(\approx\) \(0.6239871620 - 0.06919865659i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.999 + 0.0171i)T \)
3 \( 1 + (0.679 + 0.733i)T \)
5 \( 1 + (0.229 - 0.973i)T \)
7 \( 1 + (-0.967 - 0.254i)T \)
11 \( 1 + (-0.909 + 0.416i)T \)
13 \( 1 + (-0.824 - 0.565i)T \)
17 \( 1 + (-0.762 - 0.647i)T \)
19 \( 1 + (0.392 - 0.919i)T \)
23 \( 1 + (0.997 + 0.0686i)T \)
29 \( 1 + (-0.376 - 0.926i)T \)
31 \( 1 + (-0.529 - 0.848i)T \)
37 \( 1 + (0.572 - 0.819i)T \)
41 \( 1 + (0.0942 + 0.995i)T \)
43 \( 1 + (-0.992 - 0.119i)T \)
47 \( 1 + (0.128 - 0.991i)T \)
53 \( 1 + (-0.957 + 0.287i)T \)
59 \( 1 + (0.978 - 0.204i)T \)
61 \( 1 + (-0.691 + 0.722i)T \)
67 \( 1 + (-0.312 - 0.949i)T \)
71 \( 1 + (-0.529 + 0.848i)T \)
73 \( 1 + (-0.246 - 0.969i)T \)
79 \( 1 + (-0.00858 - 0.999i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.923 + 0.384i)T \)
97 \( 1 + (-0.439 - 0.898i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.3287060471715355766381927072, −24.28213408318119118651492280644, −23.4597461386215883459399950362, −22.16857251300459627368667575536, −21.28903788296261543228935025110, −20.20335077334531858498753330469, −19.296985413095673033855342193674, −18.82384707500249509366354423132, −18.204044733336486664343806501698, −17.20011014915661432988665830448, −16.08124432132508992054533207787, −15.13480443590991150359126081176, −14.36770643231170666156733395155, −13.142206773214267604012661438480, −12.33193839386621720137189431189, −11.13957201305639878636792637996, −10.16216368135046040926383117855, −9.34430119693960815482803680500, −8.40400897623196164211431036367, −7.29178821580795895227221317200, −6.74685643698981004093124954272, −5.7896873463324132291933663430, −3.35514174971961458309736414067, −2.73827969291413997121618578139, −1.72025451639508122099709622527, 0.32534230155647876371159184086, 2.235789791730058214227623586950, 3.028428596007093975750132815320, 4.58475781990471296067337640712, 5.5804061604393498824382211431, 7.14105242846257456852459894289, 7.930258212237599350021371209010, 9.10909933405843423691134373783, 9.54975272221538193729994964146, 10.30580239637062466850387124399, 11.43715159278965001266585515238, 12.86585753699677756479977769125, 13.39519850688816849370247208997, 15.0659213540695243671969118670, 15.623788221959243018907269793596, 16.44723315100368411630212104061, 17.13533860946232809219167487375, 18.19382733478468632120845529127, 19.418367563098810932823907148539, 20.02017675393863263319881991191, 20.55195866047550760376403332842, 21.45767622796828463666682916555, 22.479628098806722455761456509601, 23.79454418664817778784595899428, 24.95375623485439818644095157109

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