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Properties

Label 1-185-185.117-r0-0-0
Degree $1$
Conductor $185$
Sign $0.309 + 0.950i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
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  + 2-s + i·3-s + 4-s + i·6-s + i·7-s + 8-s − 9-s − 11-s + i·12-s + 13-s + i·14-s + 16-s − 17-s − 18-s − i·19-s + ⋯
L(s)  = 1  + 2-s + i·3-s + 4-s + i·6-s + i·7-s + 8-s − 9-s − 11-s + i·12-s + 13-s + i·14-s + 16-s − 17-s − 18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.632022581 + 1.185436015i\)
\(L(\frac12)\) \(\approx\) \(1.632022581 + 1.185436015i\)
\(L(1)\) \(\approx\) \(1.637383524 + 0.6882919030i\)
\(L(1)\) \(\approx\) \(1.637383524 + 0.6882919030i\)

Euler product

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

−26.75072038736064020680842857593, −25.770170676927291264873408825, −24.91315579314334377445084850999, −23.92499883785298600979048103892, −23.251448079165258814554635580952, −22.703838733315383532633930699782, −21.119470578354510170076228194458, −20.44507564083346419484082003441, −19.485956103063139823791063808590, −18.39796226043886493051670646172, −17.233260430279082945514532374642, −16.213694703124230435087629396940, −15.0664872964205282579052344200, −13.74388001199481144310911494500, −13.41963422183659660208322268419, −12.4378802915916927408865661539, −11.21549415102181018204792272383, −10.47289305057252004445979440103, −8.39756680658782042214861647663, −7.3821224404951073427123883112, −6.50960060648362454782247058557, −5.40352823974154834541122902594, −3.99224757851488732060259449770, −2.72922698673276380900083951834, −1.34997355114832335988066479918, 2.36880494449159151423437170477, 3.32250613779158287304168052522, 4.69754931366315092298568661469, 5.44158136636232032508847350865, 6.558753569757280301894658723697, 8.253435513622746741880674155282, 9.327587326901655809914791173959, 10.86047071103146404180716066841, 11.27789214106169524768839475049, 12.71296511855203266759140036694, 13.60364854937547757163733857495, 14.90706103716289258319360564935, 15.573647329818375593848694249374, 16.12192390220646033925562660236, 17.52691690898407662370291873364, 18.86791037614366990200049363256, 20.1844398689865297481265216076, 20.94069995245874763287118440213, 21.720541548581737407516243592107, 22.42935726000173856554614282005, 23.42230632596758788179517169609, 24.406630282190555289918637489250, 25.631572264368867810164829318112, 26.112236835561584189516754382532, 27.559681584107962079962844832421

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