Properties

Label 2-188760-1.1-c1-0-23
Degree $2$
Conductor $188760$
Sign $-1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 4·7-s + 9-s − 13-s − 15-s − 6·17-s + 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·35-s − 2·37-s + 39-s − 10·41-s + 4·43-s + 45-s + 8·47-s + 9·49-s + 6·51-s − 2·53-s + 4·59-s − 14·61-s − 4·63-s − 65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 1.45·17-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.160·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s + 0.520·59-s − 1.79·61-s − 0.503·63-s − 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.43017074385745, −12.76386204105567, −12.49961014960687, −12.00372262500330, −11.59040737666125, −10.75722245054978, −10.58473601464113, −10.13135848934364, −9.560703930579517, −9.140838371647045, −8.825477529012664, −8.120071853049097, −7.364378639506989, −6.992821430170514, −6.507165611147986, −6.124271962929566, −5.716101011898569, −5.049432544377119, −4.495205791953018, −3.962885568679200, −3.335513702732330, −2.781899499359334, −2.136374092432314, −1.583165940716244, −0.5476856088506360, 0, 0.5476856088506360, 1.583165940716244, 2.136374092432314, 2.781899499359334, 3.335513702732330, 3.962885568679200, 4.495205791953018, 5.049432544377119, 5.716101011898569, 6.124271962929566, 6.507165611147986, 6.992821430170514, 7.364378639506989, 8.120071853049097, 8.825477529012664, 9.140838371647045, 9.560703930579517, 10.13135848934364, 10.58473601464113, 10.75722245054978, 11.59040737666125, 12.00372262500330, 12.49961014960687, 12.76386204105567, 13.43017074385745

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