Properties

Label 2-166410-1.1-c1-0-18
Degree $2$
Conductor $166410$
Sign $-1$
Analytic cond. $1328.79$
Root an. cond. $36.4525$
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 − 8-s + 10-s − 4·11-s − 3·13-s + 16-s − 3·17-s − 4·19-s − 20-s + 4·22-s + 23-s + 25-s + 3·26-s − 2·29-s − 3·31-s − 32-s + 3·34-s + 3·37-s + 4·38-s + 40-s − 12·41-s − 4·44-s − 46-s − 3·47-s − 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.832·13-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.588·26-s − 0.371·29-s − 0.538·31-s − 0.176·32-s + 0.514·34-s + 0.493·37-s + 0.648·38-s + 0.158·40-s − 1.87·41-s − 0.603·44-s − 0.147·46-s − 0.437·47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(166410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 43^{2}\)
Sign: $-1$
Analytic conductor: \(1328.79\)
Root analytic conductor: \(36.4525\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{166410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 166410,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
43 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.36141876135904, −12.89118873286130, −12.55415468199635, −12.03344931084001, −11.44963126430884, −10.91146002998204, −10.79386707533931, −10.07149713813295, −9.694669067818313, −9.231888154519267, −8.484482240646284, −8.263961322985121, −7.803679231648905, −7.182935944866339, −6.841054762203567, −6.299698971655530, −5.598303214456560, −5.020107937481876, −4.652673607138982, −3.935199218314823, −3.232216937584099, −2.731395628892380, −2.088164639286206, −1.642517806920778, −0.4969536313871338, 0, 0.4969536313871338, 1.642517806920778, 2.088164639286206, 2.731395628892380, 3.232216937584099, 3.935199218314823, 4.652673607138982, 5.020107937481876, 5.598303214456560, 6.299698971655530, 6.841054762203567, 7.182935944866339, 7.803679231648905, 8.263961322985121, 8.484482240646284, 9.231888154519267, 9.694669067818313, 10.07149713813295, 10.79386707533931, 10.91146002998204, 11.44963126430884, 12.03344931084001, 12.55415468199635, 12.89118873286130, 13.36141876135904

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