↑

Properties

Label 1-372-372.215-r1-0-0
Degree $1$
Conductor $372$
Sign $-0.606 - 0.795i$
Analytic cond. $39.9769$
Root an. cond. $39.9769$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.809 + 0.587i)23-s + 25-s + (0.309 + 0.951i)29-s + (−0.809 + 0.587i)35-s − 37-s + (−0.309 − 0.951i)41-s + (0.309 + 0.951i)43-s + (0.309 − 0.951i)47-s + (0.309 − 0.951i)49-s + ⋯
L(s)  = 1  − 5-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.809 + 0.587i)23-s + 25-s + (0.309 + 0.951i)29-s + (−0.809 + 0.587i)35-s − 37-s + (−0.309 − 0.951i)41-s + (0.309 + 0.951i)43-s + (0.309 − 0.951i)47-s + (0.309 − 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(372\)    =    \(2^{2} \cdot 3 \cdot 31\)
Sign: $-0.606 - 0.795i$
Analytic conductor: \(39.9769\)
Root analytic conductor: \(39.9769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{372} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 372,\ (1:\ ),\ -0.606 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4437936817 - 0.8962349235i\)
\(L(\frac12)\) \(\approx\) \(0.4437936817 - 0.8962349235i\)
\(L(1)\) \(\approx\) \(0.8803040266 - 0.1831037694i\)
\(L(1)\) \(\approx\) \(0.8803040266 - 0.1831037694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
31 \( 1 \)
good5 \( 1 - T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.70588448038609344452409399921, −23.92060657353866894754193419505, −22.830615549960160742078717446752, −22.33996696236370654579899840748, −21.10277844420569793986631057820, −20.30096697690578835296095263581, −19.4665765580893182213654918578, −18.63910771377224165241392332000, −17.59137026341926670373429233659, −16.880677418869860461145281420605, −15.48096387061152903551927062899, −15.10509758641388318761725241069, −14.23660207795047244677998404779, −12.71969045049962810682572968754, −12.13192861950724557308641142246, −11.220841666743697517643185748199, −10.30557369718256007316646535296, −8.905430135800970282468920403617, −8.1952805063442459359208753571, −7.278771858680669625028383030561, −6.08085665910487638761572710464, −4.79993389097521263411802423533, −4.020299051703583599940438260007, −2.66794371735567726320264626697, −1.32200352028257941279553754929, 0.2982414282793041827284903226, 1.59735366066463259747267096969, 3.19948264742525803202853657069, 4.285832575117281685870731933811, 4.978800867964563610471246602264, 6.75568015788928382417093744755, 7.26439281326505011206771070523, 8.54362733158767101559496551270, 9.17910153650485855434906493248, 10.82129824298977521375820644401, 11.34368685423398473293772133954, 12.117113977762669744416785987111, 13.471948147229807248183768446215, 14.275523251709065930902622152595, 15.14911774860723659041960091585, 16.13148031025271771578169444206, 16.98255931949385328733988806997, 17.82931121291797525591199393377, 19.06238095635763282419369990179, 19.62807487430373315161130784572, 20.48449789528920835862110404621, 21.50816196319401719358342525375, 22.35618824935917582853587088854, 23.41705794716386681383791634558, 24.061298497660791397862329413108

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