Properties

Label 2-1932-1.1-c1-0-16
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 − 1.61·5-s − 7-s + 9-s − 3·11-s + 0.618·13-s − 1.61·15-s + 6.70·17-s − 4.23·19-s − 21-s − 23-s − 2.38·25-s + 27-s − 1.76·29-s − 3·31-s − 3·33-s + 1.61·35-s + 9.47·37-s + 0.618·39-s − 11.9·41-s − 9.09·43-s − 1.61·45-s − 9.23·47-s + 49-s + 6.70·51-s − 9.32·53-s + 4.85·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.723·5-s − 0.377·7-s + 0.333·9-s − 0.904·11-s + 0.171·13-s − 0.417·15-s + 1.62·17-s − 0.971·19-s − 0.218·21-s − 0.208·23-s − 0.476·25-s + 0.192·27-s − 0.327·29-s − 0.538·31-s − 0.522·33-s + 0.273·35-s + 1.55·37-s + 0.0989·39-s − 1.86·41-s − 1.38·43-s − 0.241·45-s − 1.34·47-s + 0.142·49-s + 0.939·51-s − 1.28·53-s + 0.654·55-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 \( 1 + 1.61T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 0.618T + 13T^{2} \)
17 \( 1 - 6.70T + 17T^{2} \)
19 \( 1 + 4.23T + 19T^{2} \)
29 \( 1 + 1.76T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 9.47T + 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + 9.09T + 43T^{2} \)
47 \( 1 + 9.23T + 47T^{2} \)
53 \( 1 + 9.32T + 53T^{2} \)
59 \( 1 + 9.61T + 59T^{2} \)
61 \( 1 - 5.85T + 61T^{2} \)
67 \( 1 - 6.56T + 67T^{2} \)
71 \( 1 + 4.61T + 71T^{2} \)
73 \( 1 - 1.76T + 73T^{2} \)
79 \( 1 + 6.70T + 79T^{2} \)
83 \( 1 + 7.47T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−8.558956470719536268682309433030, −8.049486843936738470746652085513, −7.48670368017283528488447982189, −6.50827852513291290224241119764, −5.56887439675100329374204240796, −4.60536603581385114101347172527, −3.60005369286613938794179311800, −3.01344737718619690504388969003, −1.71866424722829054788062165324, 0, 1.71866424722829054788062165324, 3.01344737718619690504388969003, 3.60005369286613938794179311800, 4.60536603581385114101347172527, 5.56887439675100329374204240796, 6.50827852513291290224241119764, 7.48670368017283528488447982189, 8.049486843936738470746652085513, 8.558956470719536268682309433030

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