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Properties

Label 2-2952-2952.1739-c0-0-1
Degree $2$
Conductor $2952$
Sign $0.773 - 0.633i$
Analytic cond. $1.47323$
Root an. cond. $1.21377$
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  + (−0.358 + 0.933i)2-s + (0.913 − 0.406i)3-s + (−0.743 − 0.669i)4-s + (0.0523 + 0.998i)6-s + (0.891 − 0.453i)8-s + (0.669 − 0.743i)9-s + (−0.287 + 0.602i)11-s + (−0.951 − 0.309i)12-s + (0.104 + 0.994i)16-s + (0.169 − 0.198i)17-s + (0.453 + 0.891i)18-s + (0.321 + 1.33i)19-s + (−0.459 − 0.484i)22-s + (0.629 − 0.777i)24-s + (0.994 − 0.104i)25-s + ⋯
L(s)  = 1  + (−0.358 + 0.933i)2-s + (0.913 − 0.406i)3-s + (−0.743 − 0.669i)4-s + (0.0523 + 0.998i)6-s + (0.891 − 0.453i)8-s + (0.669 − 0.743i)9-s + (−0.287 + 0.602i)11-s + (−0.951 − 0.309i)12-s + (0.104 + 0.994i)16-s + (0.169 − 0.198i)17-s + (0.453 + 0.891i)18-s + (0.321 + 1.33i)19-s + (−0.459 − 0.484i)22-s + (0.629 − 0.777i)24-s + (0.994 − 0.104i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(1.47323\)
Root analytic conductor: \(1.21377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2952,\ (\ :0),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375653937\)
\(L(\frac12)\) \(\approx\) \(1.375653937\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.358 - 0.933i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
41 \( 1 + (0.544 + 0.838i)T \)
good5 \( 1 + (-0.994 + 0.104i)T^{2} \)
7 \( 1 + (-0.544 + 0.838i)T^{2} \)
11 \( 1 + (0.287 - 0.602i)T + (-0.629 - 0.777i)T^{2} \)
13 \( 1 + (-0.838 + 0.544i)T^{2} \)
17 \( 1 + (-0.169 + 0.198i)T + (-0.156 - 0.987i)T^{2} \)
19 \( 1 + (-0.321 - 1.33i)T + (-0.891 + 0.453i)T^{2} \)
23 \( 1 + (0.978 + 0.207i)T^{2} \)
29 \( 1 + (-0.933 + 0.358i)T^{2} \)
31 \( 1 + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (-1.85 - 0.712i)T + (0.743 + 0.669i)T^{2} \)
47 \( 1 + (-0.998 + 0.0523i)T^{2} \)
53 \( 1 + (-0.156 + 0.987i)T^{2} \)
59 \( 1 + (-0.514 - 0.0541i)T + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (0.207 + 0.978i)T^{2} \)
67 \( 1 + (0.487 + 1.02i)T + (-0.629 + 0.777i)T^{2} \)
71 \( 1 + (0.987 - 0.156i)T^{2} \)
73 \( 1 + (-0.147 - 0.147i)T + iT^{2} \)
79 \( 1 + (0.965 + 0.258i)T^{2} \)
83 \( 1 + (-0.535 - 0.309i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.0819 - 0.133i)T + (-0.453 - 0.891i)T^{2} \)
97 \( 1 + (1.19 + 0.821i)T + (0.358 + 0.933i)T^{2} \)
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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

−8.840624025917929755866224850699, −8.164978728799631333872111285756, −7.53794894269517318169303359305, −7.00509712962532263742576863694, −6.16636194592206084237286764398, −5.30947391779002779041820256770, −4.36002795963090670337238113863, −3.52157413900317571331483272714, −2.29343878453337915113991364269, −1.18871372161720198984248935301, 1.14248791490428311065977906012, 2.48791409547179112261188044645, 2.97547504093640505292026304608, 3.90358073294986043720358556285, 4.68117474028701034402898684335, 5.48144813287258455282357173876, 6.89317881165319913420295345799, 7.64565898343604810873273555243, 8.353801214045532404193873132285, 9.025801876783799141758338253927

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