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Properties

Label 1-185-185.29-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.527 + 0.849i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.527 + 0.849i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.527 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9001091189 + 1.618523000i\)
\(L(\frac12)\) \(\approx\) \(0.9001091189 + 1.618523000i\)
\(L(1)\) \(\approx\) \(1.246580372 + 0.5022449738i\)
\(L(1)\) \(\approx\) \(1.246580372 + 0.5022449738i\)

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

−26.674650899383449533823169999670, −25.93805394758823496475584917146, −24.21690718591520002371389488643, −23.58603646283815127134720055926, −22.82374323793221669934310366434, −21.80132601998789744304032873312, −20.85571213703511497254838833463, −20.516740700801109363340695962421, −19.14464023449799646663543316152, −17.8583218628951159390617467771, −16.7136425958631910200448991696, −15.68973116264773678820763937020, −14.910057017840579925637173191021, −13.75155942675759684058438037465, −12.86141359918584710970183803883, −11.419175115199532462269887317874, −10.83445148495825182149396331098, −10.07621004447785353353061595681, −8.57371914296390545104543344847, −6.80903731837888384902125445134, −5.74469223382393744293106336420, −4.489893873268840211704030050169, −3.966413663329096927844190246244, −2.317829846849179921754928768603, −0.490041515095635759796740199133, 1.82530210916908036025084375453, 3.01633030727476334941834371145, 4.86807078378908708705258940359, 5.65143626870383901813775084183, 6.625472391078992633769478223785, 7.8569576337560584905775912435, 8.589699622970304610546597432318, 10.83277962423472852098057382372, 11.58756802010387456750385993681, 12.731832875986161857236428234216, 13.27008402616332906533541070901, 14.46615887224237352453885531107, 15.57443647015887677773175837385, 16.37317406682414107587522464171, 17.86492356608748792922934496983, 18.10915947458509538989275155602, 19.60904068636811643217786524746, 20.91013103990571114288619314002, 21.697653771066088202826302152747, 22.81791079014500137965187680147, 23.49695569723890434693613325096, 24.32161288386743474758577548846, 25.19279019435990661173676446203, 25.82674889349103351338854527047, 27.359934876146971814023624325963

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