Properties

Label 2-1932-1.1-c1-0-18
Degree $2$
Conductor $1932$
Sign $-1$
Analytic cond. $15.4270$
Root an. cond. $3.92773$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.38·5-s + 7-s + 9-s − 11-s − 6.09·13-s − 1.38·15-s − 5·17-s + 5.47·19-s + 21-s + 23-s − 3.09·25-s + 27-s − 1.76·29-s + 4.70·31-s − 33-s − 1.38·35-s − 7.47·37-s − 6.09·39-s − 10.2·41-s − 9.32·43-s − 1.38·45-s − 9.70·47-s + 49-s − 5·51-s + 2.61·53-s + 1.38·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.618·5-s + 0.377·7-s + 0.333·9-s − 0.301·11-s − 1.68·13-s − 0.356·15-s − 1.21·17-s + 1.25·19-s + 0.218·21-s + 0.208·23-s − 0.618·25-s + 0.192·27-s − 0.327·29-s + 0.845·31-s − 0.174·33-s − 0.233·35-s − 1.22·37-s − 0.975·39-s − 1.59·41-s − 1.42·43-s − 0.206·45-s − 1.41·47-s + 0.142·49-s − 0.700·51-s + 0.359·53-s + 0.186·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(15.4270\)
Root analytic conductor: \(3.92773\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1932,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 - T \)
good5 \( 1 + 1.38T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 6.09T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 5.47T + 19T^{2} \)
29 \( 1 + 1.76T + 29T^{2} \)
31 \( 1 - 4.70T + 31T^{2} \)
37 \( 1 + 7.47T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 9.32T + 43T^{2} \)
47 \( 1 + 9.70T + 47T^{2} \)
53 \( 1 - 2.61T + 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 8.32T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 9.47T + 73T^{2} \)
79 \( 1 - 2.70T + 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 7.76T + 97T^{2} \)
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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

−8.579704083504205905857733171891, −8.137659041066516298016612437905, −7.22186441976559038444930069588, −6.81623291539093492575427722249, −5.26427202175593157361200938295, −4.78549825442755682913731445418, −3.73277189873131724902456897366, −2.79610498753449416152967632039, −1.81395240462105523158445141432, 0, 1.81395240462105523158445141432, 2.79610498753449416152967632039, 3.73277189873131724902456897366, 4.78549825442755682913731445418, 5.26427202175593157361200938295, 6.81623291539093492575427722249, 7.22186441976559038444930069588, 8.137659041066516298016612437905, 8.579704083504205905857733171891

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