Properties

Label 2-1932-1.1-c1-0-14
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.61·5-s − 7-s + 9-s − 4.23·11-s + 2.61·13-s − 1.61·15-s + 2.23·17-s − 5.47·19-s + 21-s + 23-s − 2.38·25-s − 27-s − 3.47·29-s − 6.23·31-s + 4.23·33-s − 1.61·35-s + 3.47·37-s − 2.61·39-s − 1.47·41-s + 7.56·43-s + 1.61·45-s − 5.23·47-s + 49-s − 2.23·51-s + 4.09·53-s − 6.85·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.723·5-s − 0.377·7-s + 0.333·9-s − 1.27·11-s + 0.726·13-s − 0.417·15-s + 0.542·17-s − 1.25·19-s + 0.218·21-s + 0.208·23-s − 0.476·25-s − 0.192·27-s − 0.644·29-s − 1.12·31-s + 0.737·33-s − 0.273·35-s + 0.570·37-s − 0.419·39-s − 0.229·41-s + 1.15·43-s + 0.241·45-s − 0.763·47-s + 0.142·49-s − 0.313·51-s + 0.561·53-s − 0.924·55-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 \( 1 + 4.23T + 11T^{2} \)
13 \( 1 - 2.61T + 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 + 5.47T + 19T^{2} \)
29 \( 1 + 3.47T + 29T^{2} \)
31 \( 1 + 6.23T + 31T^{2} \)
37 \( 1 - 3.47T + 37T^{2} \)
41 \( 1 + 1.47T + 41T^{2} \)
43 \( 1 - 7.56T + 43T^{2} \)
47 \( 1 + 5.23T + 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 + 5.61T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 5.61T + 67T^{2} \)
71 \( 1 + 4.14T + 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 + 3.76T + 79T^{2} \)
83 \( 1 - 9.94T + 83T^{2} \)
89 \( 1 - 3.85T + 89T^{2} \)
97 \( 1 - 4.23T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.932526094701324319897592091112, −7.940875296716682488891264803776, −7.22458274767723424721983952468, −6.07568979709678840556042024904, −5.84197714289754961492890811542, −4.88057864523329737380107176582, −3.84059241710182488095682273370, −2.69648303044189013539176822333, −1.62857204923978028627735473441, 0, 1.62857204923978028627735473441, 2.69648303044189013539176822333, 3.84059241710182488095682273370, 4.88057864523329737380107176582, 5.84197714289754961492890811542, 6.07568979709678840556042024904, 7.22458274767723424721983952468, 7.940875296716682488891264803776, 8.932526094701324319897592091112

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