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Properties

Label 1-6300-6300.191-r0-0-0
Degree $1$
Conductor $6300$
Sign $-0.959 - 0.282i$
Analytic cond. $29.2570$
Root an. cond. $29.2570$
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  + (0.309 + 0.951i)11-s + (0.669 + 0.743i)13-s + (−0.913 + 0.406i)17-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)23-s + (−0.913 − 0.406i)29-s + (0.104 − 0.994i)31-s + (−0.978 + 0.207i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s + (−0.104 − 0.994i)47-s + (0.104 + 0.994i)53-s + (−0.978 + 0.207i)59-s + (−0.978 − 0.207i)61-s + (0.104 − 0.994i)67-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)11-s + (0.669 + 0.743i)13-s + (−0.913 + 0.406i)17-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)23-s + (−0.913 − 0.406i)29-s + (0.104 − 0.994i)31-s + (−0.978 + 0.207i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s + (−0.104 − 0.994i)47-s + (0.104 + 0.994i)53-s + (−0.978 + 0.207i)59-s + (−0.978 − 0.207i)61-s + (0.104 − 0.994i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.959 - 0.282i$
Analytic conductor: \(29.2570\)
Root analytic conductor: \(29.2570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6300,\ (0:\ ),\ -0.959 - 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01880862557 + 0.1302721090i\)
\(L(\frac12)\) \(\approx\) \(0.01880862557 + 0.1302721090i\)
\(L(1)\) \(\approx\) \(0.8924096961 + 0.1035293039i\)
\(L(1)\) \(\approx\) \(0.8924096961 + 0.1035293039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (0.104 - 0.994i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−17.28726996670828884999484654931, −16.506946710465504204544027493153, −16.043492820093636997853665215193, −15.41395428548383017540677767715, −14.55157126267131626490398971721, −14.07829634521537027068373528349, −13.30797262051899940586221000733, −12.7946683697734666149594122029, −11.98672205698967315065274795468, −11.3182505473595137912475362762, −10.62167307665515600483460136970, −10.23195946824707857831066058048, −9.00684869511857893891874966543, −8.788820772606460473230496210954, −8.00376479268205692054474368531, −7.218250458129059533343337757814, −6.40955258948238560017169796134, −5.87267946427583947669754232313, −5.11557179284534126328303392960, −4.30150522082993382957845907884, −3.40972576618558597259471746222, −3.00869213183384256736014101336, −1.868417197318687475477432338940, −1.1225963798295212910634683693, −0.03215671933994800621951719137, 1.38652597156374473466029308498, 1.930169397150863715102620273922, 2.792973525366987402541371183805, 3.82389019962530424614900022772, 4.29373610378013745410512315864, 5.069500985208357873410710461766, 5.920665976894624498969026245721, 6.674524400997738794804809202864, 7.19047537559511426422203788166, 7.92937803122474517800429592921, 8.96616651496540599539152955262, 9.19295789404167709528363844341, 9.99392885476085444526223757004, 10.90042712720019944142307350846, 11.35970254135222005142990318981, 12.03257707897566138420872244177, 12.7989762465851963733684586315, 13.56102701765547902207163946663, 13.82081607209210665298860454978, 15.0199553352154216640224670468, 15.263421980797191886640573936081, 15.88847230304960918659046309760, 16.90861602809544551730060112650, 17.22797915262070312947635291537, 17.95004436744906682671322733027

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