Properties

Label 2-280e2-1.1-c1-0-36
Degree $2$
Conductor $78400$
Sign $1$
Analytic cond. $626.027$
Root an. cond. $25.0205$
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  − 3-s − 2·9-s − 2·11-s + 4·17-s − 2·19-s − 23-s + 5·27-s − 9·29-s − 4·31-s + 2·33-s + 4·37-s − 41-s + 9·43-s − 4·51-s − 10·53-s + 2·57-s − 10·59-s + 9·61-s + 5·67-s + 69-s + 14·71-s + 12·73-s + 14·79-s + 81-s − 11·83-s + 9·87-s + 15·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.603·11-s + 0.970·17-s − 0.458·19-s − 0.208·23-s + 0.962·27-s − 1.67·29-s − 0.718·31-s + 0.348·33-s + 0.657·37-s − 0.156·41-s + 1.37·43-s − 0.560·51-s − 1.37·53-s + 0.264·57-s − 1.30·59-s + 1.15·61-s + 0.610·67-s + 0.120·69-s + 1.66·71-s + 1.40·73-s + 1.57·79-s + 1/9·81-s − 1.20·83-s + 0.964·87-s + 1.58·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(626.027\)
Root analytic conductor: \(25.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 78400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9021495554\)
\(L(\frac12)\) \(\approx\) \(0.9021495554\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.13727968081775, −13.49649214340604, −12.88824184418418, −12.46506213367149, −12.17212064785753, −11.32771646632643, −11.03685912549586, −10.76928347332439, −9.998387624279034, −9.478471202007795, −9.095394135691133, −8.327392140641612, −7.804065598807003, −7.561023054836964, −6.641369670210201, −6.263322593976194, −5.489897008515915, −5.409213774290939, −4.697132239308362, −3.852237074573940, −3.448786308949114, −2.611912621936091, −2.104393747162055, −1.176989751318968, −0.3424755097835835, 0.3424755097835835, 1.176989751318968, 2.104393747162055, 2.611912621936091, 3.448786308949114, 3.852237074573940, 4.697132239308362, 5.409213774290939, 5.489897008515915, 6.263322593976194, 6.641369670210201, 7.561023054836964, 7.804065598807003, 8.327392140641612, 9.095394135691133, 9.478471202007795, 9.998387624279034, 10.76928347332439, 11.03685912549586, 11.32771646632643, 12.17212064785753, 12.46506213367149, 12.88824184418418, 13.49649214340604, 14.13727968081775

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