Properties

Label 2-182070-1.1-c1-0-78
Degree $2$
Conductor $182070$
Sign $-1$
Analytic cond. $1453.83$
Root an. cond. $38.1292$
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  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 4·11-s − 4·13-s − 14-s + 16-s − 20-s + 4·22-s + 6·23-s + 25-s − 4·26-s − 28-s − 2·29-s + 6·31-s + 32-s + 35-s − 6·37-s − 40-s − 2·41-s + 10·43-s + 4·44-s + 6·46-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.223·20-s + 0.852·22-s + 1.25·23-s + 1/5·25-s − 0.784·26-s − 0.188·28-s − 0.371·29-s + 1.07·31-s + 0.176·32-s + 0.169·35-s − 0.986·37-s − 0.158·40-s − 0.312·41-s + 1.52·43-s + 0.603·44-s + 0.884·46-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1453.83\)
Root analytic conductor: \(38.1292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{182070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 182070,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.46713222451323, −12.74143727922697, −12.50574934550080, −11.90114339389994, −11.83324460924986, −11.01001665302369, −10.78567285488407, −10.09737705485261, −9.537219295166848, −9.205138175843486, −8.648895404454325, −8.031129097730559, −7.452430075300189, −6.969295792122830, −6.717121084298845, −6.083897823318656, −5.526295652097526, −4.901627441810557, −4.519267959775251, −3.945651019918270, −3.444480655463608, −2.854787305298474, −2.383391865938022, −1.517253521345663, −0.9215445603264761, 0, 0.9215445603264761, 1.517253521345663, 2.383391865938022, 2.854787305298474, 3.444480655463608, 3.945651019918270, 4.519267959775251, 4.901627441810557, 5.526295652097526, 6.083897823318656, 6.717121084298845, 6.969295792122830, 7.452430075300189, 8.031129097730559, 8.648895404454325, 9.205138175843486, 9.537219295166848, 10.09737705485261, 10.78567285488407, 11.01001665302369, 11.83324460924986, 11.90114339389994, 12.50574934550080, 12.74143727922697, 13.46713222451323

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