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Properties

Label 2-66248-1.1-c1-0-10
Degree $2$
Conductor $66248$
Sign $1$
Analytic cond. $528.992$
Root an. cond. $22.9998$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 3·9-s + 4·11-s + 6·17-s + 8·19-s − 25-s + 6·29-s + 8·31-s + 2·37-s + 2·41-s − 4·43-s − 6·45-s − 8·47-s + 6·53-s + 8·55-s + 6·61-s + 4·67-s + 8·71-s + 10·73-s + 16·79-s + 9·81-s + 8·83-s + 12·85-s − 6·89-s + 16·95-s − 6·97-s − 12·99-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s + 1.20·11-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 0.824·53-s + 1.07·55-s + 0.768·61-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 1.80·79-s + 81-s + 0.878·83-s + 1.30·85-s − 0.635·89-s + 1.64·95-s − 0.609·97-s − 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66248\)    =    \(2^{3} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(528.992\)
Root analytic conductor: \(22.9998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 66248,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.332750575\)
\(L(\frac12)\) \(\approx\) \(4.332750575\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.04729409603325, −13.89401224072598, −13.45208716234066, −12.60072944138722, −12.01280877364238, −11.78273092306296, −11.36562615608782, −10.58276126421749, −9.915710236220564, −9.667798425637485, −9.300867179585178, −8.499909005993596, −8.099062605594238, −7.561163261900896, −6.691717194691816, −6.377982082949123, −5.798535730260205, −5.236630518125290, −4.888114017584148, −3.827748260783434, −3.362528839593265, −2.786510221743805, −2.085066809201918, −1.132655531654130, −0.8415694056830108, 0.8415694056830108, 1.132655531654130, 2.085066809201918, 2.786510221743805, 3.362528839593265, 3.827748260783434, 4.888114017584148, 5.236630518125290, 5.798535730260205, 6.377982082949123, 6.691717194691816, 7.561163261900896, 8.099062605594238, 8.499909005993596, 9.300867179585178, 9.667798425637485, 9.915710236220564, 10.58276126421749, 11.36562615608782, 11.78273092306296, 12.01280877364238, 12.60072944138722, 13.45208716234066, 13.89401224072598, 14.04729409603325

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