Properties

Label 2-252-1.1-c3-0-0
Degree $2$
Conductor $252$
Sign $1$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·5-s − 7·7-s − 4·11-s + 54·13-s + 14·17-s + 92·19-s + 152·23-s + 71·25-s + 106·29-s − 144·31-s + 98·35-s + 158·37-s + 390·41-s − 508·43-s + 528·47-s + 49·49-s − 606·53-s + 56·55-s + 364·59-s + 678·61-s − 756·65-s + 844·67-s + 8·71-s − 422·73-s + 28·77-s + 384·79-s + 548·83-s + ⋯
L(s)  = 1  − 1.25·5-s − 0.377·7-s − 0.109·11-s + 1.15·13-s + 0.199·17-s + 1.11·19-s + 1.37·23-s + 0.567·25-s + 0.678·29-s − 0.834·31-s + 0.473·35-s + 0.702·37-s + 1.48·41-s − 1.80·43-s + 1.63·47-s + 1/7·49-s − 1.57·53-s + 0.137·55-s + 0.803·59-s + 1.42·61-s − 1.44·65-s + 1.53·67-s + 0.0133·71-s − 0.676·73-s + 0.0414·77-s + 0.546·79-s + 0.724·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.372920706\)
\(L(\frac12)\) \(\approx\) \(1.372920706\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + p T \)
good5 \( 1 + 14 T + p^{3} T^{2} \)
11 \( 1 + 4 T + p^{3} T^{2} \)
13 \( 1 - 54 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
19 \( 1 - 92 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 - 106 T + p^{3} T^{2} \)
31 \( 1 + 144 T + p^{3} T^{2} \)
37 \( 1 - 158 T + p^{3} T^{2} \)
41 \( 1 - 390 T + p^{3} T^{2} \)
43 \( 1 + 508 T + p^{3} T^{2} \)
47 \( 1 - 528 T + p^{3} T^{2} \)
53 \( 1 + 606 T + p^{3} T^{2} \)
59 \( 1 - 364 T + p^{3} T^{2} \)
61 \( 1 - 678 T + p^{3} T^{2} \)
67 \( 1 - 844 T + p^{3} T^{2} \)
71 \( 1 - 8 T + p^{3} T^{2} \)
73 \( 1 + 422 T + p^{3} T^{2} \)
79 \( 1 - 384 T + p^{3} T^{2} \)
83 \( 1 - 548 T + p^{3} T^{2} \)
89 \( 1 + 1194 T + p^{3} T^{2} \)
97 \( 1 + 1502 T + p^{3} T^{2} \)
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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

−11.48016647561398649217381205683, −10.92664613291796964048540556198, −9.619435359985852412357928789070, −8.586336500963135596512388445969, −7.68576081818539569826698250945, −6.73689945832073349827511786316, −5.38967628174797762816757374268, −4.02492502434093766009312278552, −3.09962663529711795514319748122, −0.871160386801707880068417972706, 0.871160386801707880068417972706, 3.09962663529711795514319748122, 4.02492502434093766009312278552, 5.38967628174797762816757374268, 6.73689945832073349827511786316, 7.68576081818539569826698250945, 8.586336500963135596512388445969, 9.619435359985852412357928789070, 10.92664613291796964048540556198, 11.48016647561398649217381205683

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