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Properties

Label 2-2952-328.163-c0-0-4
Degree $2$
Conductor $2952$
Sign $1$
Analytic cond. $1.47323$
Root an. cond. $1.21377$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 1.41·7-s + 8-s − 1.41·13-s + 1.41·14-s + 16-s + 25-s − 1.41·26-s + 1.41·28-s − 1.41·29-s + 32-s − 41-s − 1.41·47-s + 1.00·49-s + 50-s − 1.41·52-s + 1.41·53-s + 1.41·56-s − 1.41·58-s − 2·59-s + 64-s + 1.41·71-s − 1.41·79-s − 82-s − 2.00·91-s − 1.41·94-s + ⋯
L(s)  = 1  + 2-s + 4-s + 1.41·7-s + 8-s − 1.41·13-s + 1.41·14-s + 16-s + 25-s − 1.41·26-s + 1.41·28-s − 1.41·29-s + 32-s − 41-s − 1.41·47-s + 1.00·49-s + 50-s − 1.41·52-s + 1.41·53-s + 1.41·56-s − 1.41·58-s − 2·59-s + 64-s + 1.41·71-s − 1.41·79-s − 82-s − 2.00·91-s − 1.41·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(1.47323\)
Root analytic conductor: \(1.21377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (163, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2952,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.656960952\)
\(L(\frac12)\) \(\approx\) \(2.656960952\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
41 \( 1 + T \)
good5 \( 1 - T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.795108392049218870596471507619, −7.904563491624194386018791466300, −7.39665976595689284788762076254, −6.66852359158943879725824155862, −5.57666228184439917912274110199, −4.96834404894689526998256234754, −4.48909992216036262057439910401, −3.39917044175867866512561792433, −2.36704556648750950079930593596, −1.56862462223279559615527180303, 1.56862462223279559615527180303, 2.36704556648750950079930593596, 3.39917044175867866512561792433, 4.48909992216036262057439910401, 4.96834404894689526998256234754, 5.57666228184439917912274110199, 6.66852359158943879725824155862, 7.39665976595689284788762076254, 7.904563491624194386018791466300, 8.795108392049218870596471507619

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