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Properties

Label 1-185-185.57-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.175 - 0.984i$
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  + (0.939 − 0.342i)2-s + (0.342 − 0.939i)3-s + (0.766 − 0.642i)4-s − i·6-s + (−0.984 + 0.173i)7-s + (0.5 − 0.866i)8-s + (−0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.342 − 0.939i)12-s + (−0.766 + 0.642i)13-s + (−0.866 + 0.5i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)18-s + (0.342 − 0.939i)19-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.342 − 0.939i)3-s + (0.766 − 0.642i)4-s − i·6-s + (−0.984 + 0.173i)7-s + (0.5 − 0.866i)8-s + (−0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.342 − 0.939i)12-s + (−0.766 + 0.642i)13-s + (−0.866 + 0.5i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)18-s + (0.342 − 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.175 - 0.984i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.175 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280452847 - 1.529138790i\)
\(L(\frac12)\) \(\approx\) \(1.280452847 - 1.529138790i\)
\(L(1)\) \(\approx\) \(1.479056571 - 0.9602090167i\)
\(L(1)\) \(\approx\) \(1.479056571 - 0.9602090167i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.984 - 0.173i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.984 + 0.173i)T \)
83 \( 1 + (0.642 - 0.766i)T \)
89 \( 1 + (-0.984 - 0.173i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−27.23108722583899252226645227982, −26.41517612058004027332815770562, −25.20992003153475165038065576667, −25.0809171293260764027197266960, −23.29428297257890961087970731184, −22.55191802507888866294176891796, −22.08811197001260374133133865156, −20.6810583194588679484861063627, −20.28179980692203658096023980365, −19.121743429776411370295511454196, −17.23761912193135689889184083560, −16.53135898925457105187022832886, −15.60209937357942920284267038047, −14.76121808166762118751553413052, −13.9466851298063981066403746590, −12.70826087253474817065205485520, −11.84630333134979889481882543052, −10.3294248902212441372590539103, −9.60801488970000087003786477131, −8.067557917273879180881104290928, −6.935888390320624845093706293578, −5.64106121655468129489649042644, −4.577740781584560246682850802005, −3.51666725491470128363069749510, −2.551396635261980172410957238, 1.28007640676093998026605312798, 2.775227425919271926814158335714, 3.55393941770839534287674587232, 5.31658967836284112009029268928, 6.46266680881300101081395403478, 7.136308166300717570743399826471, 8.774338015207963149526653716252, 9.94400761761321797694891956656, 11.45921743200242880104965265350, 12.21618411045395069293029273431, 13.14488062011723899506088313521, 13.933638028965385342360577489823, 14.8279271751799291924474324150, 16.05847970621105795131163346166, 17.14554330614382501762071044426, 18.736938485770084664435865070657, 19.43980563581182984584273991924, 19.92636827690845038291050988876, 21.461660945931212443339039045804, 22.07660192203086503563258067443, 23.34029008900198029874314080205, 23.88735067665470571506163552455, 24.95410903132962601618193758124, 25.57698873129571926240514500650, 26.81224483816933999851028411280

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