Properties

Label 1-300-300.47-r1-0-0
Degree $1$
Conductor $300$
Sign $-0.929 + 0.368i$
Analytic cond. $32.2394$
Root an. cond. $32.2394$
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·7-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (−0.587 + 0.809i)23-s + (0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.587 − 0.809i)37-s + (0.809 − 0.587i)41-s − i·43-s + (−0.951 + 0.309i)47-s − 49-s + (−0.951 + 0.309i)53-s + (0.809 − 0.587i)59-s + ⋯
L(s)  = 1  − i·7-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (−0.587 + 0.809i)23-s + (0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.587 − 0.809i)37-s + (0.809 − 0.587i)41-s − i·43-s + (−0.951 + 0.309i)47-s − 49-s + (−0.951 + 0.309i)53-s + (0.809 − 0.587i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.929 + 0.368i$
Analytic conductor: \(32.2394\)
Root analytic conductor: \(32.2394\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 300,\ (1:\ ),\ -0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004449638503 + 0.02332582190i\)
\(L(\frac12)\) \(\approx\) \(0.004449638503 + 0.02332582190i\)
\(L(1)\) \(\approx\) \(0.8136348624 - 0.09409221997i\)
\(L(1)\) \(\approx\) \(0.8136348624 - 0.09409221997i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−25.599215053689380367448520180644, −24.80091421315868319060145496471, −23.952950687380342711158171381420, −22.76729846654796965454043291131, −22.24791072083349862680740971232, −20.997291409222908885449245379761, −20.452764319479548295044680965185, −19.23003452671280362770084218840, −18.26722212429478463311232873338, −17.76994684584199629323751679494, −16.38929714772051383271038831882, −15.47696083953643344837192982065, −14.90619470450471015457187561961, −13.52700371651412275547952556629, −12.691152300484598607557193781751, −11.81794739917382854939156022507, −10.64741893927676357746477521426, −9.75679440345867549346058800404, −8.526790415691698720171282043651, −7.82816929724439706922689902041, −6.33662591852845609648165877657, −5.52425037217490962647047096280, −4.32383855831339530196662677871, −2.90018709942176317149521534961, −1.87079891940733253935477891096, 0.00685129693379231336966036958, 1.442838015462525034966226529004, 2.976331465127971784524552078422, 4.12423998949052918504673080695, 5.19557342746413390245716643264, 6.55005026853404217505833268084, 7.38818775226179277703403546752, 8.55348363010768169607552719464, 9.55172167319532722939549637285, 10.832304303832183584368696754925, 11.26968538408521758680382869688, 12.769400175714313364449573729882, 13.651942607224203503787972085285, 14.2422263243377918268777626086, 15.84396957982165127195915791578, 16.1611493599641590302773075788, 17.50105485781558625477561931528, 18.14558565797446576896300498330, 19.37438167505163167272785542905, 20.065358558445397077017904681099, 21.09623679894797758107627708869, 21.84125391732182235658963892954, 23.023610987970251867180336338180, 23.79159260367648669444012593884, 24.38576163373852091211335973130

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