Properties

Label 2-700-1.1-c1-0-0
Degree $2$
Conductor $700$
Sign $1$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  − 2·3-s + 7-s + 9-s + 3·11-s − 4·13-s + 2·19-s − 2·21-s − 3·23-s + 4·27-s + 9·29-s + 8·31-s − 6·33-s + 5·37-s + 8·39-s − 6·41-s + 11·43-s + 6·47-s + 49-s + 6·53-s − 4·57-s − 10·61-s + 63-s + 5·67-s + 6·69-s + 15·71-s − 10·73-s + 3·77-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s + 0.458·19-s − 0.436·21-s − 0.625·23-s + 0.769·27-s + 1.67·29-s + 1.43·31-s − 1.04·33-s + 0.821·37-s + 1.28·39-s − 0.937·41-s + 1.67·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.529·57-s − 1.28·61-s + 0.125·63-s + 0.610·67-s + 0.722·69-s + 1.78·71-s − 1.17·73-s + 0.341·77-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: $1$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.016943091\)
\(L(\frac12)\) \(\approx\) \(1.016943091\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−10.48340708300327334608204066535, −9.811649660309562675106236923024, −8.758337549295479378590072229628, −7.73562448419965250225099053149, −6.73034779603349979037904446310, −6.01118714823649537133179937027, −5.01328815310058208778255110773, −4.26504108499817032327826623435, −2.63651692432818486959856402458, −0.925578357219550016990910217681, 0.925578357219550016990910217681, 2.63651692432818486959856402458, 4.26504108499817032327826623435, 5.01328815310058208778255110773, 6.01118714823649537133179937027, 6.73034779603349979037904446310, 7.73562448419965250225099053149, 8.758337549295479378590072229628, 9.811649660309562675106236923024, 10.48340708300327334608204066535

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