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Properties

Label 1-185-185.63-r1-0-0
Degree $1$
Conductor $185$
Sign $0.644 + 0.764i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)9-s + 11-s + (0.866 + 0.5i)12-s + (0.866 − 0.5i)13-s − 14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)9-s + 11-s + (0.866 + 0.5i)12-s + (0.866 − 0.5i)13-s − 14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.644 + 0.764i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.644 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.875992115 + 1.801404158i\)
\(L(\frac12)\) \(\approx\) \(3.875992115 + 1.801404158i\)
\(L(1)\) \(\approx\) \(2.272492680 + 0.6408059560i\)
\(L(1)\) \(\approx\) \(2.272492680 + 0.6408059560i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 - T \)
31 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 - iT \)
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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

−26.79946702661941151262149677047, −25.7754978819631957054914622379, −24.96705955159777559605389967001, −23.90868652277994579812713154829, −22.704699135636709030392380340805, −22.12873053819101222180299920903, −20.95402642002816193238180711624, −20.299211503241279928607447108023, −19.41745997611607047823260453202, −18.68370945299884975111557083407, −16.64911719043803901919544880160, −15.92988545111724298012316185896, −14.8301941283994161144045961130, −13.89895896128479385301374689105, −13.31308705604730998822071712718, −12.02682165581028253916214873626, −10.84541208231910676706980063454, −9.797470919408957046045482185658, −9.01383298558352160423664736529, −7.26060744371695802824067612867, −6.19622229551611807440014522972, −4.608193522282537551929049323528, −3.69515324847073128294684559911, −2.82049594341687468815554555004, −1.20134797481881114648601274575, 1.61993516509753482632350990938, 3.24072786253104252002218783458, 3.738541196627590531902666385335, 5.70046445253553867008226375149, 6.50177132890227714893185318017, 7.67715961773922533101882710214, 8.65229193918472378844959175673, 9.8056813014266473185788402915, 11.67700407535228486058130936984, 12.53748475326376151029584332346, 13.37659127331363198395426256591, 14.28795454086290985936878119973, 15.19603730965151302936457658557, 16.06209896741638570017752220385, 17.2309593898328085161842863875, 18.54092253368550393821695231752, 19.50224155168371676819341327674, 20.48730816343648327731117840560, 21.397588002544612846044779263856, 22.510551196936095943057988204723, 23.28905461877097229934251683314, 24.41177815890740547204473743702, 25.303106059733753203150855030530, 25.61795081194035479807986037581, 26.72082975511269388788276523433

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