Properties

Label 1-344-344.189-r0-0-0
Degree $1$
Conductor $344$
Sign $-0.492 - 0.870i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (−0.826 + 0.563i)3-s + (0.733 + 0.680i)5-s + (−0.5 − 0.866i)7-s + (0.365 − 0.930i)9-s + (−0.623 − 0.781i)11-s + (−0.955 + 0.294i)13-s + (−0.988 − 0.149i)15-s + (−0.733 + 0.680i)17-s + (−0.365 − 0.930i)19-s + (0.900 + 0.433i)21-s + (−0.988 + 0.149i)23-s + (0.0747 + 0.997i)25-s + (0.222 + 0.974i)27-s + (−0.826 − 0.563i)29-s + (0.0747 − 0.997i)31-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (0.733 + 0.680i)5-s + (−0.5 − 0.866i)7-s + (0.365 − 0.930i)9-s + (−0.623 − 0.781i)11-s + (−0.955 + 0.294i)13-s + (−0.988 − 0.149i)15-s + (−0.733 + 0.680i)17-s + (−0.365 − 0.930i)19-s + (0.900 + 0.433i)21-s + (−0.988 + 0.149i)23-s + (0.0747 + 0.997i)25-s + (0.222 + 0.974i)27-s + (−0.826 − 0.563i)29-s + (0.0747 − 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ -0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1510138683 - 0.2589281919i\)
\(L(\frac12)\) \(\approx\) \(0.1510138683 - 0.2589281919i\)
\(L(1)\) \(\approx\) \(0.6263385038 + 0.01378151074i\)
\(L(1)\) \(\approx\) \(0.6263385038 + 0.01378151074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.955 + 0.294i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (-0.365 - 0.930i)T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + (0.0747 - 0.997i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (-0.365 - 0.930i)T \)
71 \( 1 + (-0.988 - 0.149i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.826 + 0.563i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−25.12344652865848825257990204739, −24.34855899090814053692423750057, −23.51210063328751819408132441627, −22.2948111650244484032891692595, −22.03795696261019107923714975722, −20.77785878520426521385020784237, −19.88406091643697088709382749973, −18.68814802632406407967458932771, −18.024740029298124457113831772762, −17.26032941154984401060750420947, −16.34264025511243188451709344924, −15.5319504678975792494459493449, −14.21201941374090875930710668607, −13.01332557484807883673424963921, −12.54344325057162070008370084689, −11.814605779963228806351313636356, −10.34967529485455632456036989274, −9.6841551913334514267094357698, −8.47830976737474555093786651668, −7.31211106030520644273624735311, −6.23658736471289883408067979248, −5.389029637183985354968910027566, −4.65120407807222775067610095122, −2.56354693180171883366794711871, −1.72602484999985127672761232775, 0.18964295437051251245570066701, 2.194296085587478282903251910921, 3.53434155106344196401432116248, 4.59564250068534429406041629070, 5.83603161987506009870957989108, 6.520379193967482428410393476116, 7.54571487760066871547474979744, 9.21922983163921576009506540546, 10.04884782725027320322900341496, 10.73462499915940780503746066800, 11.49901506846886513054149213495, 12.926560785491187376897327508960, 13.62300033538105147177849338352, 14.78599068720395643119994840307, 15.66483311649463710260492707542, 16.76953650169333057079073472453, 17.26506749813880720091425779361, 18.187513456985587048706149863314, 19.20685610887132439908828531980, 20.25516219924766132565187462170, 21.43469810237153986951267943036, 21.940047326221530543224877547953, 22.63654250015851224471905421770, 23.71461421301225856326751145379, 24.29158528325307439013326563146

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