Properties

Degree $2$
Conductor $700$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 3·9-s − 5·11-s − 6·13-s + 4·17-s − 6·19-s + 3·23-s − 3·29-s + 2·31-s + 7·37-s − 4·41-s − 7·43-s − 2·47-s + 49-s − 10·53-s − 14·59-s + 4·61-s − 3·63-s + 3·67-s − 13·71-s + 16·73-s − 5·77-s + 79-s + 9·81-s + 10·83-s + 10·89-s − 6·91-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s − 1.50·11-s − 1.66·13-s + 0.970·17-s − 1.37·19-s + 0.625·23-s − 0.557·29-s + 0.359·31-s + 1.15·37-s − 0.624·41-s − 1.06·43-s − 0.291·47-s + 1/7·49-s − 1.37·53-s − 1.82·59-s + 0.512·61-s − 0.377·63-s + 0.366·67-s − 1.54·71-s + 1.87·73-s − 0.569·77-s + 0.112·79-s + 81-s + 1.09·83-s + 1.05·89-s − 0.628·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{700} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.96588467417808, −19.15077126603132, −18.58752888079085, −17.64672974215136, −16.99206826878377, −16.55270986003725, −15.28157847784093, −14.85838135616716, −14.17043158979118, −13.15031445833060, −12.50638115913623, −11.66776369482124, −10.80057867668664, −10.11841877195286, −9.180615470968494, −8.046653320309692, −7.708695425430839, −6.413912222923018, −5.303123946152911, −4.766299899149961, −3.108715382715318, −2.243486812089597, 0, 2.243486812089597, 3.108715382715318, 4.766299899149961, 5.303123946152911, 6.413912222923018, 7.708695425430839, 8.046653320309692, 9.180615470968494, 10.11841877195286, 10.80057867668664, 11.66776369482124, 12.50638115913623, 13.15031445833060, 14.17043158979118, 14.85838135616716, 15.28157847784093, 16.55270986003725, 16.99206826878377, 17.64672974215136, 18.58752888079085, 19.15077126603132, 19.96588467417808

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