Properties

Label 2-84150-1.1-c1-0-91
Degree $2$
Conductor $84150$
Sign $1$
Analytic cond. $671.941$
Root an. cond. $25.9218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4·7-s + 8-s + 11-s + 4·13-s + 4·14-s + 16-s − 17-s − 8·19-s + 22-s + 4·26-s + 4·28-s + 10·31-s + 32-s − 34-s − 8·37-s − 8·38-s + 10·41-s + 8·43-s + 44-s + 10·47-s + 9·49-s + 4·52-s − 12·53-s + 4·56-s + 8·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 0.301·11-s + 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 1.83·19-s + 0.213·22-s + 0.784·26-s + 0.755·28-s + 1.79·31-s + 0.176·32-s − 0.171·34-s − 1.31·37-s − 1.29·38-s + 1.56·41-s + 1.21·43-s + 0.150·44-s + 1.45·47-s + 9/7·49-s + 0.554·52-s − 1.64·53-s + 0.534·56-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(671.941\)
Root analytic conductor: \(25.9218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{84150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 84150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.298050853\)
\(L(\frac12)\) \(\approx\) \(6.298050853\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.08318809995047, −13.48689394873091, −13.05493296490957, −12.28204224425806, −12.17808197678890, −11.32130770777492, −11.10883522473501, −10.61756723314375, −10.27469702260035, −9.280914549838412, −8.729694147763658, −8.427136289810193, −7.840403676627961, −7.356861539107877, −6.581190575126426, −6.141001241010341, −5.762420003813348, −4.867467666785550, −4.554956520248872, −4.074097237760455, −3.514679412404317, −2.530852743807459, −2.125199518616422, −1.403492082294809, −0.7490044958208349, 0.7490044958208349, 1.403492082294809, 2.125199518616422, 2.530852743807459, 3.514679412404317, 4.074097237760455, 4.554956520248872, 4.867467666785550, 5.762420003813348, 6.141001241010341, 6.581190575126426, 7.356861539107877, 7.840403676627961, 8.427136289810193, 8.729694147763658, 9.280914549838412, 10.27469702260035, 10.61756723314375, 11.10883522473501, 11.32130770777492, 12.17808197678890, 12.28204224425806, 13.05493296490957, 13.48689394873091, 14.08318809995047

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