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Properties

Label 1-185-185.122-r1-0-0
Degree $1$
Conductor $185$
Sign $0.198 + 0.980i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.198 + 0.980i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.198 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.238243963 + 1.831138070i\)
\(L(\frac12)\) \(\approx\) \(2.238243963 + 1.831138070i\)
\(L(1)\) \(\approx\) \(1.501978524 + 0.7483835226i\)
\(L(1)\) \(\approx\) \(1.501978524 + 0.7483835226i\)

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

−27.32281900662375982888423785151, −25.221382137390274608502784519788, −24.72058048680533901800636947687, −23.62301868815417475694907688236, −23.052857614841200812054415041350, −21.978645573889873882215699682439, −21.27508204786781930583496280830, −20.224589040873840966190277657047, −18.84206908507100839460020112034, −18.39790542321426128716098420465, −16.95870403197070419821642292506, −16.03825918500611435882949007608, −14.64400868882229305945267821348, −13.952496332354216780559341806132, −12.6342202086205632662129917672, −11.86620357492887365389718567523, −11.218405597261150135881722993565, −10.108094626332796198464172504457, −8.48667054496972422061596356073, −6.91480424932734597969741156298, −6.0010933432649148526665804000, −5.01796860735105733323917703295, −3.87684862260649256554256694774, −2.0507145517865155508687562333, −1.08035261087651637561186401962, 1.297718374306428975627183868118, 3.51669746751939606575021283735, 4.387916069142653963728688741997, 5.465709899422676054639002439107, 6.417698938884036188268668873616, 7.56635238032292339164654592992, 8.89216925793736090468234466803, 10.56806599306147389866384746293, 11.363066565911077019724466089661, 12.24595654575834956773148221510, 13.46108774891595029134792688889, 14.56659978791914971659464460250, 15.37506781628692354871859246053, 16.450339078284707121639188930169, 17.27111712064292068638186771055, 17.92786203179007666047739570352, 19.78335177257309704786037644734, 20.9368654658386072595076355181, 21.54289083932236548907627390870, 22.50646309569330640239633237919, 23.46895579723650307332396112256, 23.90744123741338912102220335026, 25.16207199569473619058254956962, 26.114565794133712394746680211686, 27.3569905217488432936800962835

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