Properties

Label 2-70e2-1.1-c1-0-26
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s + 3·11-s + 4·13-s − 2·19-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 + 6·53-s − 4·57-s + 10·61-s + 5·67-s − 6·69-s + 15·71-s + 10·73-s − 7·79-s − 11·81-s − 12·83-s + 18·87-s + 12·89-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s + 0.904·11-s + 1.10·13-s − 0.458·19-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 + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.610·67-s − 0.722·69-s + 1.78·71-s + 1.17·73-s − 0.787·79-s − 1.22·81-s − 1.31·83-s + 1.92·87-s + 1.27·89-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: yes
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.315751859\)
\(L(\frac12)\) \(\approx\) \(3.315751859\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
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

−8.302892491342254910347877193939, −7.80026429092781153881633790359, −6.82104727985831383123636504801, −6.21179142960385677297493895179, −5.40673316579866536330795337231, −4.08308162740428403937139621322, −3.87618020070886168399849825036, −2.84294326033506919310791275790, −2.06563936265242068894460467465, −0.987839877051285196558457230133, 0.987839877051285196558457230133, 2.06563936265242068894460467465, 2.84294326033506919310791275790, 3.87618020070886168399849825036, 4.08308162740428403937139621322, 5.40673316579866536330795337231, 6.21179142960385677297493895179, 6.82104727985831383123636504801, 7.80026429092781153881633790359, 8.302892491342254910347877193939

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