Properties

Label 2-70e2-1.1-c1-0-31
Degree $2$
Conductor $4900$
Sign $1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s + 5.00·9-s + 4·11-s − 4.24·13-s − 1.41·17-s + 2.82·19-s + 4·23-s + 5.65·27-s + 8·29-s + 11.3·33-s + 8·37-s − 12·39-s − 7.07·41-s + 4·43-s − 5.65·47-s − 4.00·51-s − 10·53-s + 8.00·57-s + 14.1·59-s − 7.07·61-s + 11.3·69-s + 7.07·73-s + 8·79-s + 1.00·81-s + 14.1·83-s + 22.6·87-s + 7.07·89-s + ⋯
L(s)  = 1  + 1.63·3-s + 1.66·9-s + 1.20·11-s − 1.17·13-s − 0.342·17-s + 0.648·19-s + 0.834·23-s + 1.08·27-s + 1.48·29-s + 1.96·33-s + 1.31·37-s − 1.92·39-s − 1.10·41-s + 0.609·43-s − 0.825·47-s − 0.560·51-s − 1.37·53-s + 1.05·57-s + 1.84·59-s − 0.905·61-s + 1.36·69-s + 0.827·73-s + 0.900·79-s + 0.111·81-s + 1.55·83-s + 2.42·87-s + 0.749·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.995993141\)
\(L(\frac12)\) \(\approx\) \(3.995993141\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 3T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7.07T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 - 1.41T + 97T^{2} \)
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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

−8.261230630956351170838363690348, −7.71045004352119243779776988851, −6.93481390856791769066453092985, −6.41393421431243039060607194330, −5.06056133590690184033807746277, −4.42885631005143267058009024336, −3.55209071954943527049885074619, −2.88028168055179038414671515265, −2.12964667996718740327044579309, −1.07768887130158783799120547121, 1.07768887130158783799120547121, 2.12964667996718740327044579309, 2.88028168055179038414671515265, 3.55209071954943527049885074619, 4.42885631005143267058009024336, 5.06056133590690184033807746277, 6.41393421431243039060607194330, 6.93481390856791769066453092985, 7.71045004352119243779776988851, 8.261230630956351170838363690348

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