Properties

Label 2-3920-1.1-c1-0-38
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
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  + 2.56·3-s − 5-s + 3.56·9-s + 2.56·11-s + 5.68·13-s − 2.56·15-s − 3.43·17-s + 1.12·19-s + 5.12·23-s + 25-s + 1.43·27-s + 4.56·29-s − 10.2·31-s + 6.56·33-s + 8.24·37-s + 14.5·39-s − 7.12·41-s − 1.12·43-s − 3.56·45-s + 6.56·47-s − 8.80·51-s − 4.87·53-s − 2.56·55-s + 2.87·57-s − 4·59-s + 15.1·61-s − 5.68·65-s + ⋯
L(s)  = 1  + 1.47·3-s − 0.447·5-s + 1.18·9-s + 0.772·11-s + 1.57·13-s − 0.661·15-s − 0.833·17-s + 0.257·19-s + 1.06·23-s + 0.200·25-s + 0.276·27-s + 0.847·29-s − 1.84·31-s + 1.14·33-s + 1.35·37-s + 2.33·39-s − 1.11·41-s − 0.171·43-s − 0.530·45-s + 0.957·47-s − 1.23·51-s − 0.669·53-s − 0.345·55-s + 0.381·57-s − 0.520·59-s + 1.93·61-s − 0.705·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.489888499\)
\(L(\frac12)\) \(\approx\) \(3.489888499\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good3 \( 1 - 2.56T + 3T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 - 5.68T + 13T^{2} \)
17 \( 1 + 3.43T + 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 + 1.12T + 43T^{2} \)
47 \( 1 - 6.56T + 47T^{2} \)
53 \( 1 + 4.87T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 3.12T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.439552931393957149076411583155, −8.024683287841735942732738858122, −7.04125185857772899732359736917, −6.54851986778516274802624188173, −5.45371551071751576857553904097, −4.30969940657618854871906577743, −3.71631234761095866487642495601, −3.12164619535452430905706479218, −2.08676654064415276949547323493, −1.07702418655814939321757275903, 1.07702418655814939321757275903, 2.08676654064415276949547323493, 3.12164619535452430905706479218, 3.71631234761095866487642495601, 4.30969940657618854871906577743, 5.45371551071751576857553904097, 6.54851986778516274802624188173, 7.04125185857772899732359736917, 8.024683287841735942732738858122, 8.439552931393957149076411583155

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