Properties

Label 2-1932-1.1-c1-0-13
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 − 1.30·5-s + 7-s + 9-s + 3.60·11-s − 4.30·13-s + 1.30·15-s − 3·17-s − 0.394·19-s − 21-s − 23-s − 3.30·25-s − 27-s + 29-s + 31-s − 3.60·33-s − 1.30·35-s + 6.21·37-s + 4.30·39-s + 1.60·41-s + 2.90·43-s − 1.30·45-s − 3.39·47-s + 49-s + 3·51-s + 2.09·53-s − 4.69·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.582·5-s + 0.377·7-s + 0.333·9-s + 1.08·11-s − 1.19·13-s + 0.336·15-s − 0.727·17-s − 0.0904·19-s − 0.218·21-s − 0.208·23-s − 0.660·25-s − 0.192·27-s + 0.185·29-s + 0.179·31-s − 0.627·33-s − 0.220·35-s + 1.02·37-s + 0.688·39-s + 0.250·41-s + 0.443·43-s − 0.194·45-s − 0.495·47-s + 0.142·49-s + 0.420·51-s + 0.287·53-s − 0.633·55-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 \( 1 + 1.30T + 5T^{2} \)
11 \( 1 - 3.60T + 11T^{2} \)
13 \( 1 + 4.30T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 0.394T + 19T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 6.21T + 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 - 2.90T + 43T^{2} \)
47 \( 1 + 3.39T + 47T^{2} \)
53 \( 1 - 2.09T + 53T^{2} \)
59 \( 1 - 0.0916T + 59T^{2} \)
61 \( 1 + 9.51T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 1.09T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 7.60T + 83T^{2} \)
89 \( 1 + 0.302T + 89T^{2} \)
97 \( 1 + 11T + 97T^{2} \)
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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

−8.865573702542904813041700199964, −7.87823164910431246022748568392, −7.25276574617372903731106419117, −6.44940168022507556725179244512, −5.62332677581800205276268885908, −4.49217842734020754909329312001, −4.16807982310039301498395542834, −2.76289467099899513060619439256, −1.50373751912312532066113730990, 0, 1.50373751912312532066113730990, 2.76289467099899513060619439256, 4.16807982310039301498395542834, 4.49217842734020754909329312001, 5.62332677581800205276268885908, 6.44940168022507556725179244512, 7.25276574617372903731106419117, 7.87823164910431246022748568392, 8.865573702542904813041700199964

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