Properties

Label 2-1932-1.1-c1-0-12
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 + 0.236·11-s + 0.381·13-s + 0.618·15-s − 2.23·17-s + 3.47·19-s + 21-s + 23-s − 4.61·25-s − 27-s + 5.47·29-s − 1.76·31-s − 0.236·33-s + 0.618·35-s − 5.47·37-s − 0.381·39-s + 7.47·41-s − 12.5·43-s − 0.618·45-s − 0.763·47-s + 49-s + 2.23·51-s − 7.09·53-s − 0.145·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.276·5-s − 0.377·7-s + 0.333·9-s + 0.0711·11-s + 0.105·13-s + 0.159·15-s − 0.542·17-s + 0.796·19-s + 0.218·21-s + 0.208·23-s − 0.923·25-s − 0.192·27-s + 1.01·29-s − 0.316·31-s − 0.0410·33-s + 0.104·35-s − 0.899·37-s − 0.0611·39-s + 1.16·41-s − 1.91·43-s − 0.0921·45-s − 0.111·47-s + 0.142·49-s + 0.313·51-s − 0.973·53-s − 0.0196·55-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 \( 1 - 0.236T + 11T^{2} \)
13 \( 1 - 0.381T + 13T^{2} \)
17 \( 1 + 2.23T + 17T^{2} \)
19 \( 1 - 3.47T + 19T^{2} \)
29 \( 1 - 5.47T + 29T^{2} \)
31 \( 1 + 1.76T + 31T^{2} \)
37 \( 1 + 5.47T + 37T^{2} \)
41 \( 1 - 7.47T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 + 0.763T + 47T^{2} \)
53 \( 1 + 7.09T + 53T^{2} \)
59 \( 1 + 3.38T + 59T^{2} \)
61 \( 1 - 5.32T + 61T^{2} \)
67 \( 1 + 3.38T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 - 1.94T + 73T^{2} \)
79 \( 1 + 8.23T + 79T^{2} \)
83 \( 1 + 7.94T + 83T^{2} \)
89 \( 1 + 2.85T + 89T^{2} \)
97 \( 1 + 0.236T + 97T^{2} \)
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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

−8.835997115331228201459127588499, −7.976278093522204236014832747718, −7.12560688888952564741987005992, −6.45173771275297721777090503583, −5.61287756590038162270844640024, −4.76293495531927825566473534545, −3.85544166182386658974120333312, −2.87585854646542638452068144525, −1.47619492191824786851912018750, 0, 1.47619492191824786851912018750, 2.87585854646542638452068144525, 3.85544166182386658974120333312, 4.76293495531927825566473534545, 5.61287756590038162270844640024, 6.45173771275297721777090503583, 7.12560688888952564741987005992, 7.976278093522204236014832747718, 8.835997115331228201459127588499

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