Properties

Label 1-367-367.122-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.848 + 0.529i$
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  + (0.514 + 0.857i)2-s + (0.679 + 0.733i)3-s + (−0.469 + 0.882i)4-s + (0.229 − 0.973i)5-s + (−0.279 + 0.960i)6-s + (−0.967 − 0.254i)7-s + (−0.998 + 0.0514i)8-s + (−0.0771 + 0.997i)9-s + (0.952 − 0.304i)10-s + (0.815 + 0.579i)11-s + (−0.967 + 0.254i)12-s + (−0.0771 + 0.997i)13-s + (−0.279 − 0.960i)14-s + (0.870 − 0.492i)15-s + (−0.558 − 0.829i)16-s + (−0.179 + 0.983i)17-s + ⋯
L(s)  = 1  + (0.514 + 0.857i)2-s + (0.679 + 0.733i)3-s + (−0.469 + 0.882i)4-s + (0.229 − 0.973i)5-s + (−0.279 + 0.960i)6-s + (−0.967 − 0.254i)7-s + (−0.998 + 0.0514i)8-s + (−0.0771 + 0.997i)9-s + (0.952 − 0.304i)10-s + (0.815 + 0.579i)11-s + (−0.967 + 0.254i)12-s + (−0.0771 + 0.997i)13-s + (−0.279 − 0.960i)14-s + (0.870 − 0.492i)15-s + (−0.558 − 0.829i)16-s + (−0.179 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.848 + 0.529i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.848 + 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4744654992 + 1.656767089i\)
\(L(\frac12)\) \(\approx\) \(0.4744654992 + 1.656767089i\)
\(L(1)\) \(\approx\) \(1.027518089 + 1.020258958i\)
\(L(1)\) \(\approx\) \(1.027518089 + 1.020258958i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.514 + 0.857i)T \)
3 \( 1 + (0.679 + 0.733i)T \)
5 \( 1 + (0.229 - 0.973i)T \)
7 \( 1 + (-0.967 - 0.254i)T \)
11 \( 1 + (0.815 + 0.579i)T \)
13 \( 1 + (-0.0771 + 0.997i)T \)
17 \( 1 + (-0.179 + 0.983i)T \)
19 \( 1 + (0.600 + 0.799i)T \)
23 \( 1 + (-0.558 + 0.829i)T \)
29 \( 1 + (-0.376 - 0.926i)T \)
31 \( 1 + (-0.469 + 0.882i)T \)
37 \( 1 + (0.423 + 0.905i)T \)
41 \( 1 + (0.815 - 0.579i)T \)
43 \( 1 + (0.600 - 0.799i)T \)
47 \( 1 + (0.128 - 0.991i)T \)
53 \( 1 + (0.229 - 0.973i)T \)
59 \( 1 + (0.978 - 0.204i)T \)
61 \( 1 + (-0.279 - 0.960i)T \)
67 \( 1 + (0.978 + 0.204i)T \)
71 \( 1 + (-0.469 - 0.882i)T \)
73 \( 1 + (-0.716 + 0.697i)T \)
79 \( 1 + (0.870 + 0.492i)T \)
83 \( 1 + T \)
89 \( 1 + (0.128 - 0.991i)T \)
97 \( 1 + (-0.558 + 0.829i)T \)
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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

−24.38650523823298032907794846706, −23.15450171762318034082692958845, −22.38470372680936539678581161286, −21.95885813468869422157554869830, −20.59708026153657512116697992902, −19.81599150122932930111672880252, −19.2322155932506618587840497399, −18.33345066923370182728440354376, −17.83887356098739180522773829453, −16.03878751043095124547499187337, −14.92777381772865634299880305598, −14.2700988386861138932857682827, −13.44041789329535903338457847484, −12.729585941545068435497918598281, −11.72783380676161003842733071599, −10.80371970837754572473425889540, −9.58388282624349278646442699643, −9.074904084536351580814724076067, −7.48449780021103268916959781736, −6.449587109606120172745807486959, −5.73489269842029376530151854531, −3.898738701356058932499423807019, −2.93172624987202126160872123347, −2.50944947603938391710151838426, −0.82408912704779244570517451244, 1.96614967577633061198748190380, 3.74019139865665407583317295656, 4.049003238705906165741328732078, 5.27272504983253854801170496735, 6.3050312346459049004666427604, 7.469483348034331567361796755775, 8.55398538702432354537258830953, 9.359969873895428787000665394277, 9.9338696313497522652760117409, 11.788643763464286224058752288631, 12.700365504149925871929965590161, 13.60775187525782608450981230694, 14.265879209801474028024851646594, 15.31432510829614868640533749620, 16.14741738113787485308009920173, 16.718902250336440032967570849098, 17.45307450973219807874716938077, 19.07358146212190009322969079837, 19.93581626303947678339901417922, 20.77822694809793025659813783636, 21.689059123905230841607481448320, 22.30126639174600105952375577879, 23.346851285083067654422237174348, 24.28815757134208626891009117074, 25.154595346257483252961029059265

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