Properties

Label 2-70e2-1.1-c1-0-36
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  + 2.41·3-s + 2.82·9-s + 1.82·11-s + 6.41·13-s + 3.58·17-s + 7.65·19-s − 3.41·23-s − 0.414·27-s − 4.65·29-s − 7.41·31-s + 4.41·33-s + 0.585·37-s + 15.4·39-s − 3.41·41-s − 0.343·43-s + 10.8·47-s + 8.65·51-s + 12.2·53-s + 18.4·57-s − 0.585·59-s − 10.8·61-s − 3.07·67-s − 8.24·69-s − 10.4·71-s + 10.8·73-s − 15.1·79-s − 9.48·81-s + ⋯
L(s)  = 1  + 1.39·3-s + 0.942·9-s + 0.551·11-s + 1.77·13-s + 0.869·17-s + 1.75·19-s − 0.711·23-s − 0.0797·27-s − 0.864·29-s − 1.33·31-s + 0.768·33-s + 0.0963·37-s + 2.47·39-s − 0.533·41-s − 0.0523·43-s + 1.58·47-s + 1.21·51-s + 1.68·53-s + 2.44·57-s − 0.0762·59-s − 1.38·61-s − 0.375·67-s − 0.992·69-s − 1.24·71-s + 1.26·73-s − 1.70·79-s − 1.05·81-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: no
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.963074439\)
\(L(\frac12)\) \(\approx\) \(3.963074439\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 3T^{2} \)
11 \( 1 - 1.82T + 11T^{2} \)
13 \( 1 - 6.41T + 13T^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
23 \( 1 + 3.41T + 23T^{2} \)
29 \( 1 + 4.65T + 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 - 0.585T + 37T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 + 0.343T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 0.585T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 3.07T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + 9.72T + 97T^{2} \)
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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

−8.361251205470429203853807829585, −7.56714521937640808791754525658, −7.18736704822174484613294152477, −5.92971230632405879527100337486, −5.55664483963845982611713316925, −4.15304975336719562120775089236, −3.56787507644199132378158165433, −3.11357139442858685877803954465, −1.90588883351041407079731985295, −1.13248615436590951685514244052, 1.13248615436590951685514244052, 1.90588883351041407079731985295, 3.11357139442858685877803954465, 3.56787507644199132378158165433, 4.15304975336719562120775089236, 5.55664483963845982611713316925, 5.92971230632405879527100337486, 7.18736704822174484613294152477, 7.56714521937640808791754525658, 8.361251205470429203853807829585

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