Properties

Label 2-70e2-1.1-c1-0-64
Degree $2$
Conductor $4900$
Sign $-1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6·9-s − 5·11-s − 3·13-s − 17-s − 6·19-s − 6·23-s + 9·27-s − 9·29-s + 4·31-s − 15·33-s − 2·37-s − 9·39-s + 4·41-s − 10·43-s − 47-s − 3·51-s − 4·53-s − 18·57-s + 8·59-s + 8·61-s − 12·67-s − 18·69-s + 8·71-s + 2·73-s + 13·79-s + 9·81-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s − 1.50·11-s − 0.832·13-s − 0.242·17-s − 1.37·19-s − 1.25·23-s + 1.73·27-s − 1.67·29-s + 0.718·31-s − 2.61·33-s − 0.328·37-s − 1.44·39-s + 0.624·41-s − 1.52·43-s − 0.145·47-s − 0.420·51-s − 0.549·53-s − 2.38·57-s + 1.04·59-s + 1.02·61-s − 1.46·67-s − 2.16·69-s + 0.949·71-s + 0.234·73-s + 1.46·79-s + 81-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.139418361854724874623097588421, −7.44099969728554811267307044223, −6.73090257323593090666183314764, −5.65281322329375920967173457432, −4.73855020505789086897581548645, −4.01006359968701985097827443222, −3.18951693575181046626973642778, −2.29908472484109817747562649632, −1.98802964785301257384067880121, 0, 1.98802964785301257384067880121, 2.29908472484109817747562649632, 3.18951693575181046626973642778, 4.01006359968701985097827443222, 4.73855020505789086897581548645, 5.65281322329375920967173457432, 6.73090257323593090666183314764, 7.44099969728554811267307044223, 8.139418361854724874623097588421

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