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Properties

Label 1-367-367.118-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

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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: $\chi_{367} (118, \cdot )$
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 \( 1 + (-0.439 + 0.898i)T \)
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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

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

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