Properties

Label 2-2952-41.40-c1-0-39
Degree $2$
Conductor $2952$
Sign $0.0582 + 0.998i$
Analytic cond. $23.5718$
Root an. cond. $4.85508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.60·5-s + 0.181i·7-s − 2.05i·11-s + 0.949i·13-s − 3.83i·17-s − 6.15i·19-s + 4.73·23-s − 2.42·25-s − 3.26i·29-s − 7.12·31-s + 0.291i·35-s − 4.42·37-s + (0.372 + 6.39i)41-s − 7.54·43-s − 10.2i·47-s + ⋯
L(s)  = 1  + 0.717·5-s + 0.0687i·7-s − 0.618i·11-s + 0.263i·13-s − 0.930i·17-s − 1.41i·19-s + 0.987·23-s − 0.484·25-s − 0.605i·29-s − 1.28·31-s + 0.0493i·35-s − 0.727·37-s + (0.0582 + 0.998i)41-s − 1.15·43-s − 1.49i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0582 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0582 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $0.0582 + 0.998i$
Analytic conductor: \(23.5718\)
Root analytic conductor: \(4.85508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (2377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2952,\ (\ :1/2),\ 0.0582 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.689830502\)
\(L(\frac12)\) \(\approx\) \(1.689830502\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 + (-0.372 - 6.39i)T \)
good5 \( 1 - 1.60T + 5T^{2} \)
7 \( 1 - 0.181iT - 7T^{2} \)
11 \( 1 + 2.05iT - 11T^{2} \)
13 \( 1 - 0.949iT - 13T^{2} \)
17 \( 1 + 3.83iT - 17T^{2} \)
19 \( 1 + 6.15iT - 19T^{2} \)
23 \( 1 - 4.73T + 23T^{2} \)
29 \( 1 + 3.26iT - 29T^{2} \)
31 \( 1 + 7.12T + 31T^{2} \)
37 \( 1 + 4.42T + 37T^{2} \)
43 \( 1 + 7.54T + 43T^{2} \)
47 \( 1 + 10.2iT - 47T^{2} \)
53 \( 1 + 0.554iT - 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + 9.80T + 61T^{2} \)
67 \( 1 + 7.91iT - 67T^{2} \)
71 \( 1 + 3.73iT - 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 8.84iT - 79T^{2} \)
83 \( 1 - 1.70T + 83T^{2} \)
89 \( 1 + 4.76iT - 89T^{2} \)
97 \( 1 - 9.39iT - 97T^{2} \)
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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

−8.850463946504120442320849357879, −7.76628801802387251352102866586, −6.95953422222405964965002127268, −6.36977612389958824043747592158, −5.35658736002669903451218926146, −4.92911506209472018646943843468, −3.69990777061748047402783835077, −2.79364610578420736463928388320, −1.89260644021547095338657760799, −0.52066281085968050218160781695, 1.40268589954349397974121017253, 2.14981895577315241833913710910, 3.39042608811034907437630557455, 4.14137429206870761199712459216, 5.30763948230355957160423278126, 5.73829288538697141505940267409, 6.67613592817155349022040418563, 7.38821520147804814871590982405, 8.207073894096243505073139921294, 8.968720457608307240040229669649

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