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Properties

Label 1-204-204.11-r1-0-0
Degree $1$
Conductor $204$
Sign $-0.968 + 0.250i$
Analytic cond. $21.9228$
Root an. cond. $21.9228$
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  + (−0.382 − 0.923i)5-s + (−0.382 + 0.923i)7-s + (0.923 + 0.382i)11-s − i·13-s + (−0.707 − 0.707i)19-s + (−0.923 − 0.382i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)29-s + (−0.923 + 0.382i)31-s + 35-s + (−0.923 + 0.382i)37-s + (−0.382 + 0.923i)41-s + (−0.707 + 0.707i)43-s − i·47-s + (−0.707 − 0.707i)49-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)5-s + (−0.382 + 0.923i)7-s + (0.923 + 0.382i)11-s − i·13-s + (−0.707 − 0.707i)19-s + (−0.923 − 0.382i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)29-s + (−0.923 + 0.382i)31-s + 35-s + (−0.923 + 0.382i)37-s + (−0.382 + 0.923i)41-s + (−0.707 + 0.707i)43-s − i·47-s + (−0.707 − 0.707i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 204 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 204 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(204\)    =    \(2^{2} \cdot 3 \cdot 17\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(21.9228\)
Root analytic conductor: \(21.9228\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{204} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 204,\ (1:\ ),\ -0.968 + 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0001237130066 + 0.0009727769961i\)
\(L(\frac12)\) \(\approx\) \(0.0001237130066 + 0.0009727769961i\)
\(L(1)\) \(\approx\) \(0.7666476924 - 0.05890816286i\)
\(L(1)\) \(\approx\) \(0.7666476924 - 0.05890816286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.382 - 0.923i)T \)
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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

−26.26294342825998695306836719368, −25.418130509045646913622115562984, −24.07178127903934422912856594473, −23.31639244813086747289009517050, −22.42520440033098472263443880167, −21.605285734238283816842362242358, −20.32873972409241180433973413155, −19.32274426103297847496399929013, −18.803541994454198672665635565040, −17.407434905384912367277009722820, −16.594436680855891402932321315919, −15.5417780561591632823874045353, −14.25434092514607255529585780706, −13.87072383156998072132105481173, −12.29999460115093369299447757431, −11.32021865389123810060731000255, −10.41261371642396492488873907012, −9.35809450750062866428235210622, −7.945969604557256988646978857322, −6.88313250762703840249030481445, −6.145726409180496330963337385012, −4.162024504002703122752044603021, −3.5456812682755320588633760658, −1.84856690160665725088484094217, −0.00032364596297262507020679326, 1.63497018236444250599127821675, 3.20999887082504636084547447784, 4.54796281309247422835877647000, 5.609122534777443794009330196048, 6.82488095451198318079552955268, 8.30712812531828953697717551802, 8.96603402021482797089610760656, 10.09332631975682105842495165404, 11.56902057479519828287367157023, 12.41662368979067931478516548807, 13.0943947005451416495620563284, 14.62564495325651094819451751280, 15.5052386588439564993341151810, 16.37874278032539665175409349325, 17.41190728443721375617631845572, 18.412210683850333839039978015192, 19.731257152688911880590757213872, 20.08250526478891499216134429448, 21.445909117543535814358950213760, 22.256587915178827035284779003098, 23.26928546562874295684929911252, 24.34991021524618991989058026812, 25.08488106131974818616608903431, 25.84572401463443306869946848814, 27.38279389690009637989293027008

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