Properties

Label 2-1932-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4.30·5-s − 7-s + 9-s − 1.60·11-s − 1.30·13-s + 4.30·15-s − 7.60·17-s − 7.60·19-s + 21-s − 23-s + 13.5·25-s − 27-s + 1.60·29-s + 3.60·31-s + 1.60·33-s + 4.30·35-s − 1.60·37-s + 1.30·39-s − 3.60·41-s + 0.697·43-s − 4.30·45-s + 2.60·47-s + 49-s + 7.60·51-s − 4.90·53-s + 6.90·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.92·5-s − 0.377·7-s + 0.333·9-s − 0.484·11-s − 0.361·13-s + 1.11·15-s − 1.84·17-s − 1.74·19-s + 0.218·21-s − 0.208·23-s + 2.70·25-s − 0.192·27-s + 0.298·29-s + 0.647·31-s + 0.279·33-s + 0.727·35-s − 0.263·37-s + 0.208·39-s − 0.563·41-s + 0.106·43-s − 0.641·45-s + 0.380·47-s + 0.142·49-s + 1.06·51-s − 0.674·53-s + 0.931·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2940342573\)
\(L(\frac12)\) \(\approx\) \(0.2940342573\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.30T + 5T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 + 7.60T + 19T^{2} \)
29 \( 1 - 1.60T + 29T^{2} \)
31 \( 1 - 3.60T + 31T^{2} \)
37 \( 1 + 1.60T + 37T^{2} \)
41 \( 1 + 3.60T + 41T^{2} \)
43 \( 1 - 0.697T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 + 4.90T + 53T^{2} \)
59 \( 1 + 6.90T + 59T^{2} \)
61 \( 1 + 5.90T + 61T^{2} \)
67 \( 1 - 9.69T + 67T^{2} \)
71 \( 1 - 2.09T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 - 1.78T + 83T^{2} \)
89 \( 1 - 2.51T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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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

−8.933137885356858032540212937031, −8.401575175771316064170692889889, −7.60766984145757659779817334744, −6.81708587306179222425004814557, −6.26691375538174925191032995529, −4.75506945257044346815865649897, −4.44572679667145837816937068077, −3.50548322087512858493652973024, −2.32318375918765043319228948734, −0.35108444683536224189146360872, 0.35108444683536224189146360872, 2.32318375918765043319228948734, 3.50548322087512858493652973024, 4.44572679667145837816937068077, 4.75506945257044346815865649897, 6.26691375538174925191032995529, 6.81708587306179222425004814557, 7.60766984145757659779817334744, 8.401575175771316064170692889889, 8.933137885356858032540212937031

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