Properties

Label 2-1932-1.1-c1-0-19
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 + 0.618·5-s − 7-s + 9-s − 3·11-s − 1.61·13-s + 0.618·15-s − 6.70·17-s + 0.236·19-s − 21-s − 23-s − 4.61·25-s + 27-s − 6.23·29-s − 3·31-s − 3·33-s − 0.618·35-s + 0.527·37-s − 1.61·39-s + 5.94·41-s + 2.09·43-s + 0.618·45-s − 4.76·47-s + 49-s − 6.70·51-s + 6.32·53-s − 1.85·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.276·5-s − 0.377·7-s + 0.333·9-s − 0.904·11-s − 0.448·13-s + 0.159·15-s − 1.62·17-s + 0.0541·19-s − 0.218·21-s − 0.208·23-s − 0.923·25-s + 0.192·27-s − 1.15·29-s − 0.538·31-s − 0.522·33-s − 0.104·35-s + 0.0867·37-s − 0.259·39-s + 0.928·41-s + 0.318·43-s + 0.0921·45-s − 0.694·47-s + 0.142·49-s − 0.939·51-s + 0.868·53-s − 0.250·55-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 \( 1 - 0.618T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 + 6.70T + 17T^{2} \)
19 \( 1 - 0.236T + 19T^{2} \)
29 \( 1 + 6.23T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 0.527T + 37T^{2} \)
41 \( 1 - 5.94T + 41T^{2} \)
43 \( 1 - 2.09T + 43T^{2} \)
47 \( 1 + 4.76T + 47T^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + 7.38T + 59T^{2} \)
61 \( 1 + 0.854T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 2.38T + 71T^{2} \)
73 \( 1 - 6.23T + 73T^{2} \)
79 \( 1 - 6.70T + 79T^{2} \)
83 \( 1 - 1.47T + 83T^{2} \)
89 \( 1 - 7.79T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−9.015146915657015209417224531927, −7.929392683743312878378190862746, −7.40100983509761279854631995234, −6.45474482146570766627574299893, −5.61881713229407850963289442402, −4.64351345236886376271401914771, −3.75763484064554852632384062720, −2.64326968259702339137451269125, −1.94770729116318433966066273669, 0, 1.94770729116318433966066273669, 2.64326968259702339137451269125, 3.75763484064554852632384062720, 4.64351345236886376271401914771, 5.61881713229407850963289442402, 6.45474482146570766627574299893, 7.40100983509761279854631995234, 7.929392683743312878378190862746, 9.015146915657015209417224531927

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