Properties

Label 1-367-367.107-r0-0-0
Degree $1$
Conductor $367$
Sign $0.243 - 0.969i$
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  + (−0.179 + 0.983i)2-s + (−0.716 − 0.697i)3-s + (−0.935 − 0.352i)4-s + (−0.0771 − 0.997i)5-s + (0.815 − 0.579i)6-s + (0.423 − 0.905i)7-s + (0.514 − 0.857i)8-s + (0.0257 + 0.999i)9-s + (0.994 + 0.102i)10-s + (0.978 − 0.204i)11-s + (0.423 + 0.905i)12-s + (0.0257 + 0.999i)13-s + (0.815 + 0.579i)14-s + (−0.640 + 0.767i)15-s + (0.751 + 0.660i)16-s + (−0.894 − 0.447i)17-s + ⋯
L(s)  = 1  + (−0.179 + 0.983i)2-s + (−0.716 − 0.697i)3-s + (−0.935 − 0.352i)4-s + (−0.0771 − 0.997i)5-s + (0.815 − 0.579i)6-s + (0.423 − 0.905i)7-s + (0.514 − 0.857i)8-s + (0.0257 + 0.999i)9-s + (0.994 + 0.102i)10-s + (0.978 − 0.204i)11-s + (0.423 + 0.905i)12-s + (0.0257 + 0.999i)13-s + (0.815 + 0.579i)14-s + (−0.640 + 0.767i)15-s + (0.751 + 0.660i)16-s + (−0.894 − 0.447i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.243 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6161322409 - 0.4804284005i\)
\(L(\frac12)\) \(\approx\) \(0.6161322409 - 0.4804284005i\)
\(L(1)\) \(\approx\) \(0.7415376785 - 0.1108941878i\)
\(L(1)\) \(\approx\) \(0.7415376785 - 0.1108941878i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.179 + 0.983i)T \)
3 \( 1 + (-0.716 - 0.697i)T \)
5 \( 1 + (-0.0771 - 0.997i)T \)
7 \( 1 + (0.423 - 0.905i)T \)
11 \( 1 + (0.978 - 0.204i)T \)
13 \( 1 + (0.0257 + 0.999i)T \)
17 \( 1 + (-0.894 - 0.447i)T \)
19 \( 1 + (0.952 - 0.304i)T \)
23 \( 1 + (0.751 - 0.660i)T \)
29 \( 1 + (0.128 - 0.991i)T \)
31 \( 1 + (-0.935 - 0.352i)T \)
37 \( 1 + (-0.784 - 0.620i)T \)
41 \( 1 + (0.978 + 0.204i)T \)
43 \( 1 + (0.952 + 0.304i)T \)
47 \( 1 + (-0.843 + 0.536i)T \)
53 \( 1 + (-0.0771 - 0.997i)T \)
59 \( 1 + (-0.558 + 0.829i)T \)
61 \( 1 + (0.815 + 0.579i)T \)
67 \( 1 + (-0.558 - 0.829i)T \)
71 \( 1 + (-0.935 + 0.352i)T \)
73 \( 1 + (-0.967 - 0.254i)T \)
79 \( 1 + (-0.640 - 0.767i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.843 + 0.536i)T \)
97 \( 1 + (0.751 - 0.660i)T \)
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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

−24.972876371117755717430756592556, −23.58975511423173572786980494637, −22.55436022825425944322213036451, −22.2086790233226449740124539447, −21.58606615113731509246593479220, −20.5367831427910833576457521987, −19.64713206638333693682344112416, −18.58321820929251590583727290067, −17.7885856740825476918733732181, −17.372606417548883441903904473566, −15.88057319266064471924583145912, −14.96408089162319609503886166213, −14.25688920096084345149260331782, −12.77147427190675444382392161166, −11.88489504304583519840165551311, −11.187547348887151528492593707635, −10.53202456368221718011942743766, −9.50375111433685981412543373665, −8.72296048319562260532367225626, −7.267924638509347906575196758114, −5.90781970347993168077593403446, −5.00660642236503797185430665793, −3.751013603584518718113010755923, −2.95598009217368655783995526971, −1.496283098866069989624912264075, 0.63117052191450136359086021531, 1.5918473756276580130797095702, 4.19637617002615704371331251350, 4.75965205658624290430368408971, 5.917405931291321854502107667935, 6.91311771866988448542472099200, 7.56896889244749101660201692525, 8.7306840357834326640986983740, 9.50477041381310492404612852723, 11.00026112206290257018062041266, 11.83436971712467512041459112425, 13.04608740649707883035938848453, 13.69329112369216358163356308567, 14.503782830742892108562694329546, 16.11282873748779090078465755680, 16.48479145442451508599459927590, 17.36791788146161749223298942136, 17.841110328410237852032744977793, 19.132430835091242889832168687640, 19.778659729444042536441021165782, 21.025885316361285238261571956108, 22.306775873374817811169131168033, 22.95932367200079268288232231752, 23.95731587999966646199305271285, 24.41165276498891046881623188088

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