Properties

Label 1-195-195.164-r0-0-0
Degree $1$
Conductor $195$
Sign $-0.957 + 0.289i$
Analytic cond. $0.905576$
Root an. cond. $0.905576$
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  + i·2-s − 4-s + i·7-s i·8-s + i·11-s − 14-s + 16-s − 17-s + i·19-s − 22-s − 23-s i·28-s − 29-s + i·31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + i·7-s i·8-s + i·11-s − 14-s + 16-s − 17-s + i·19-s − 22-s − 23-s i·28-s − 29-s + i·31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ -0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1183735598 + 0.7994500432i\)
\(L(\frac12)\) \(\approx\) \(0.1183735598 + 0.7994500432i\)
\(L(1)\) \(\approx\) \(0.6142942290 + 0.5916149716i\)
\(L(1)\) \(\approx\) \(0.6142942290 + 0.5916149716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 - iT \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 \)
47 \( 1 - T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 \)
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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

−26.53970163819884286869601109403, −26.20205873205330436406302720953, −24.339907653884565881275096629752, −23.69323459756635844169686410293, −22.50706789048203771526511897538, −21.81756224779155642958134917089, −20.71809362880821641175602833471, −19.92366635228757798276734725896, −19.15838172825543823491155155481, −17.9981227256340156220179718837, −17.17064299636546259835292021488, −16.038298898148479165301523950196, −14.51746027543506212886247753767, −13.56754764567687374460655584569, −12.95255242439687484454914273737, −11.41268235140230869157901562984, −10.93198526329444982405328910280, −9.7516689282910421104382374259, −8.72051968651867390605263620428, −7.530822725444980359488024397958, −5.98584635364490389199387102701, −4.52782902890317638678781266719, −3.62244400894268124842017547758, −2.25593206268667393946227195212, −0.63009750646815891430198310668, 2.04126972779369719503925112940, 3.86657144658667160273343369276, 5.074291092357082985754028130683, 6.06963261297330422930417170921, 7.16140482539580902988353550661, 8.31270946578051809490057019791, 9.224087961347064058563092687740, 10.256544325800496919111887148415, 11.96825796772416932360022795304, 12.7875809908972514332308420575, 13.99932041331792509897496875416, 15.03312500245828279683798262155, 15.6430610809627454211536287917, 16.72519199332928960599423635461, 17.8641124960387946142430584313, 18.414810051807352470480567833, 19.60489772663022005346137826126, 20.9098484236809140232681137912, 22.12933784535157338354543743263, 22.65538483279558990330550147423, 23.84147662800606836924583005109, 24.69185414663387701958618302826, 25.46974604796932136141303764925, 26.23200205503814086363158546972, 27.41006243987611155963671540654

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