Properties

Label 1-195-195.44-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: $\chi_{195} (44, \cdot )$
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

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

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