Properties

Label 2-252-1.1-c5-0-5
Degree $2$
Conductor $252$
Sign $1$
Analytic cond. $40.4167$
Root an. cond. $6.35741$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 96·5-s + 49·7-s + 720·11-s + 572·13-s − 1.25e3·17-s − 94·19-s − 96·23-s + 6.09e3·25-s + 4.37e3·29-s − 6.24e3·31-s + 4.70e3·35-s − 1.07e4·37-s − 1.20e4·41-s − 9.16e3·43-s + 2.58e4·47-s + 2.40e3·49-s − 1.01e3·53-s + 6.91e4·55-s − 1.24e3·59-s + 7.59e3·61-s + 5.49e4·65-s + 4.11e4·67-s + 3.76e4·71-s − 1.34e4·73-s + 3.52e4·77-s + 6.24e3·79-s + 2.52e4·83-s + ⋯
L(s)  = 1  + 1.71·5-s + 0.377·7-s + 1.79·11-s + 0.938·13-s − 1.05·17-s − 0.0597·19-s − 0.0378·23-s + 1.94·25-s + 0.965·29-s − 1.16·31-s + 0.649·35-s − 1.29·37-s − 1.11·41-s − 0.755·43-s + 1.70·47-s + 1/7·49-s − 0.0495·53-s + 3.08·55-s − 0.0464·59-s + 0.261·61-s + 1.61·65-s + 1.11·67-s + 0.885·71-s − 0.295·73-s + 0.678·77-s + 0.112·79-s + 0.402·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(40.4167\)
Root analytic conductor: \(6.35741\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.417232203\)
\(L(\frac12)\) \(\approx\) \(3.417232203\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
good5 \( 1 - 96 T + p^{5} T^{2} \)
11 \( 1 - 720 T + p^{5} T^{2} \)
13 \( 1 - 44 p T + p^{5} T^{2} \)
17 \( 1 + 1254 T + p^{5} T^{2} \)
19 \( 1 + 94 T + p^{5} T^{2} \)
23 \( 1 + 96 T + p^{5} T^{2} \)
29 \( 1 - 4374 T + p^{5} T^{2} \)
31 \( 1 + 6244 T + p^{5} T^{2} \)
37 \( 1 + 10798 T + p^{5} T^{2} \)
41 \( 1 + 12006 T + p^{5} T^{2} \)
43 \( 1 + 9160 T + p^{5} T^{2} \)
47 \( 1 - 25836 T + p^{5} T^{2} \)
53 \( 1 + 1014 T + p^{5} T^{2} \)
59 \( 1 + 1242 T + p^{5} T^{2} \)
61 \( 1 - 7592 T + p^{5} T^{2} \)
67 \( 1 - 41132 T + p^{5} T^{2} \)
71 \( 1 - 37632 T + p^{5} T^{2} \)
73 \( 1 + 13438 T + p^{5} T^{2} \)
79 \( 1 - 6248 T + p^{5} T^{2} \)
83 \( 1 - 25254 T + p^{5} T^{2} \)
89 \( 1 - 45126 T + p^{5} T^{2} \)
97 \( 1 - 107222 T + p^{5} T^{2} \)
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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

−11.07997769004327497873864175886, −10.19905114445934005235328822324, −9.113586937523096138048897295155, −8.708036719053210425503338138769, −6.80923408949690372420803922762, −6.25776464124503790928344988874, −5.13878585087194986800969731221, −3.76771266103547017919565726086, −2.07592691154418043550096265874, −1.24359954126291028944787993055, 1.24359954126291028944787993055, 2.07592691154418043550096265874, 3.76771266103547017919565726086, 5.13878585087194986800969731221, 6.25776464124503790928344988874, 6.80923408949690372420803922762, 8.708036719053210425503338138769, 9.113586937523096138048897295155, 10.19905114445934005235328822324, 11.07997769004327497873864175886

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