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Properties

Label 1-367-367.181-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.844 + 0.535i$
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  + (0.941 + 0.336i)2-s + (−0.716 + 0.697i)3-s + (0.773 + 0.633i)4-s + (−0.0771 + 0.997i)5-s + (−0.909 + 0.416i)6-s + (0.423 + 0.905i)7-s + (0.514 + 0.857i)8-s + (0.0257 − 0.999i)9-s + (−0.408 + 0.912i)10-s + (−0.666 + 0.745i)11-s + (−0.996 + 0.0857i)12-s + (0.852 + 0.522i)13-s + (0.0942 + 0.995i)14-s + (−0.640 − 0.767i)15-s + (0.196 + 0.980i)16-s + (0.0600 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.941 + 0.336i)2-s + (−0.716 + 0.697i)3-s + (0.773 + 0.633i)4-s + (−0.0771 + 0.997i)5-s + (−0.909 + 0.416i)6-s + (0.423 + 0.905i)7-s + (0.514 + 0.857i)8-s + (0.0257 − 0.999i)9-s + (−0.408 + 0.912i)10-s + (−0.666 + 0.745i)11-s + (−0.996 + 0.0857i)12-s + (0.852 + 0.522i)13-s + (0.0942 + 0.995i)14-s + (−0.640 − 0.767i)15-s + (0.196 + 0.980i)16-s + (0.0600 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.844 + 0.535i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5071544930 + 1.748780190i\)
\(L(\frac12)\) \(\approx\) \(0.5071544930 + 1.748780190i\)
\(L(1)\) \(\approx\) \(1.074932942 + 1.040026834i\)
\(L(1)\) \(\approx\) \(1.074932942 + 1.040026834i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.941 + 0.336i)T \)
3 \( 1 + (-0.716 + 0.697i)T \)
5 \( 1 + (-0.0771 + 0.997i)T \)
7 \( 1 + (0.423 + 0.905i)T \)
11 \( 1 + (-0.666 + 0.745i)T \)
13 \( 1 + (0.852 + 0.522i)T \)
17 \( 1 + (0.0600 - 0.998i)T \)
19 \( 1 + (-0.212 - 0.977i)T \)
23 \( 1 + (0.196 - 0.980i)T \)
29 \( 1 + (0.128 + 0.991i)T \)
31 \( 1 + (0.162 - 0.986i)T \)
37 \( 1 + (0.929 + 0.368i)T \)
41 \( 1 + (-0.312 + 0.949i)T \)
43 \( 1 + (-0.739 - 0.672i)T \)
47 \( 1 + (-0.843 - 0.536i)T \)
53 \( 1 + (0.902 - 0.431i)T \)
59 \( 1 + (-0.558 - 0.829i)T \)
61 \( 1 + (-0.909 - 0.416i)T \)
67 \( 1 + (0.997 + 0.0686i)T \)
71 \( 1 + (0.162 + 0.986i)T \)
73 \( 1 + (0.262 - 0.964i)T \)
79 \( 1 + (0.985 + 0.170i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.0429 + 0.999i)T \)
97 \( 1 + (-0.947 + 0.320i)T \)
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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

−24.07894927246308012182334346359, −23.28893113651225455562450870396, −23.06408943554455772647273960638, −21.46025284578255296301992197159, −21.07724106627158270923336550617, −19.96664301732800916873607618362, −19.28654683612462624877771302382, −18.13460677698857428492865176632, −17.026082511846749369012794376571, −16.368718109538586318702074634, −15.44711595977306783461691839686, −13.99065192904182560592858742900, −13.31668667026444810047693824029, −12.76311217184573698827802781572, −11.749977429507883969404186904513, −10.90897799073334644809177456051, −10.1956914780592147136744212567, −8.285856346222356896002277562542, −7.603864172411051483775711455469, −6.15660486450413607397650720838, −5.56337860258271055441690387676, −4.53379937071165895893577364408, −3.4986114513421654254927519059, −1.72991280862487839280110596272, −0.93929433680560266910953784100, 2.26306102746032724981792432648, 3.213149487579140976540377694601, 4.51283081764911928717075470054, 5.198042382237412421932477596288, 6.324165822906285490800005176466, 6.97115154400024987187564610482, 8.3329675744822499223446182701, 9.66852561693529722257978826043, 10.93504278278309255643071581370, 11.39073628802528528072911172857, 12.2758682927157477973520032120, 13.42984921746782946528799565210, 14.62564955286629558907309019118, 15.19702148658566380809394027628, 15.835081148190394787404764104126, 16.785657816715164830077276901679, 18.07617485850973672436464764007, 18.416188358127903885171743873732, 20.18344923848363744216619881999, 21.03850014473399631780123021713, 21.74617369778005317688340326655, 22.39135690703705671034698463726, 23.193574878942535604356361494370, 23.77222102009257192648772309987, 24.983719001287090170179271778816

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