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Properties

Label 1-185-185.14-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.751 + 0.660i$
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 + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s − i·6-s + (0.5 − 0.866i)7-s + i·8-s + (−0.5 − 0.866i)9-s − 11-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s − i·6-s + (0.5 − 0.866i)7-s + i·8-s + (−0.5 − 0.866i)9-s − 11-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.866 + 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.751 + 0.660i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.751 + 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2111509126 + 0.5602309438i\)
\(L(\frac12)\) \(\approx\) \(0.2111509126 + 0.5602309438i\)
\(L(1)\) \(\approx\) \(0.5192689671 + 0.2410718148i\)
\(L(1)\) \(\approx\) \(0.5192689671 + 0.2410718148i\)

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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + iT \)
29 \( 1 + iT \)
31 \( 1 + iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 - T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−26.65728119848216061882998748267, −25.74346180266276378386607969431, −24.62868246379508934530526146351, −24.122441689672360352471056311541, −22.667768622817933468655766851814, −21.67680454567230011182986480693, −20.791724487963237110288386099490, −19.52748585770733205403195208577, −18.69259362484087845704583108777, −18.12739315500535262763922258456, −17.15749289692987893112313064099, −16.243300309605454283235643619430, −14.93288127103248726345735736723, −13.41002539261098354902843244505, −12.3317022512837839053006572675, −11.75163110380171473560087405715, −10.69885985589518460301519538604, −9.500866742343337103654526327684, −8.15685101314802800816177874338, −7.56171419272647666021920467682, −6.1981790552583496866443453332, −4.91656255932835176610908733090, −2.76416154437764369466726816411, −1.8990531021933293129159102872, −0.35036964689769778621081409131, 1.07640298806854312236130748427, 3.13255396471413268729603358383, 4.92950342771700650721872287148, 5.55193552370089649007297123199, 7.19406058628921000689152337316, 7.96637461590062719405766610470, 9.45386495682402592841433631683, 10.23961179597457592306339291177, 10.95070497238277332376429457150, 12.135151687098587904891119847155, 13.97793575802533620959370788117, 14.85524446041454664939523980101, 15.92452448594475365048093413748, 16.60023567615190693283936070091, 17.591583109130148375235077467172, 18.22330956017463065862471659943, 19.73655037429677292925961670682, 20.534116702028319043337572019468, 21.392732095024155166734113963092, 22.89613055172104587055365693414, 23.49544319986717950793839680920, 24.52653567795884446946248401052, 25.75363494726249033646635571630, 26.59350598300178296262550500290, 27.284742228134925759736321647462

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