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Properties

Label 1-195-195.74-r1-0-0
Degree $1$
Conductor $195$
Sign $0.964 - 0.265i$
Analytic cond. $20.9556$
Root an. cond. $20.9556$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)11-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)28-s + (0.5 + 0.866i)29-s + 31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)11-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)28-s + (0.5 + 0.866i)29-s + 31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(20.9556\)
Root analytic conductor: \(20.9556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (1:\ ),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346982519 - 0.1817286464i\)
\(L(\frac12)\) \(\approx\) \(1.346982519 - 0.1817286464i\)
\(L(1)\) \(\approx\) \(0.8677220206 - 0.2384789074i\)
\(L(1)\) \(\approx\) \(0.8677220206 - 0.2384789074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−26.80932576663522970609694086192, −25.82682584835831748514065399095, −24.74878926565600207750966935116, −24.38002251504859538998219465035, −23.21622963791602506703150062370, −22.14970933111061731014413743758, −21.25951792985862653013038627639, −19.75203281061055328887446541119, −18.99675145421078857556335310321, −17.966709537824563764466430739484, −17.2752070100030995083856455856, −16.00725860142107359157216391674, −15.38491471639385375770788722880, −14.25842597764057057965690261935, −13.44428116082480196228913404369, −11.828006779168837047883643422090, −10.897124178096585252626377625405, −9.428311602844191707442078153614, −8.741498895423645560953980180767, −7.705473708090287532585581149475, −6.42751382057441999706794491071, −5.522527205434046881041681167890, −4.333836552201062671827625973006, −2.37976870475282287167587698013, −0.70358561607020500285978017251, 1.06971400771167782877952987779, 2.21581560538708524390793784613, 3.87842189861923556281865079426, 4.601811596764155454015780048938, 6.59302952262783317371084281971, 7.80030463995093904767886833612, 8.713402239679334467337042691521, 10.07218101270789376493465056742, 10.64894844395901217101134508579, 11.87373572788350808190492340603, 12.73147296165934270066605513468, 13.869739985949295796825192734666, 14.87236774946049210998184876563, 16.48539850273925674485858795934, 17.28567463598355148823251566750, 18.03347913446071486048307351173, 19.21383451286426631882150960022, 20.119483061374471880211191301057, 20.735064955342708871661584635, 21.82576599152973196159393765848, 22.78254163542695821031577863115, 23.71623309149131889464018513955, 25.05267631918915924166002490921, 26.04878138436440135862577686842, 26.89162161199193262712583505614

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