Properties

Label 2-361998-1.1-c1-0-40
Degree $2$
Conductor $361998$
Sign $1$
Analytic cond. $2890.56$
Root an. cond. $53.7640$
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 − 2·5-s + 7-s + 8-s − 2·10-s + 14-s + 16-s − 17-s − 2·20-s + 8·23-s − 25-s + 28-s + 6·29-s + 8·31-s + 32-s − 34-s − 2·35-s − 10·37-s − 2·40-s − 6·41-s + 12·43-s + 8·46-s + 49-s − 50-s + 10·53-s + 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s − 0.338·35-s − 1.64·37-s − 0.316·40-s − 0.937·41-s + 1.82·43-s + 1.17·46-s + 1/7·49-s − 0.141·50-s + 1.37·53-s + 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(361998\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(2890.56\)
Root analytic conductor: \(53.7640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{361998} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 361998,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.346738374\)
\(L(\frac12)\) \(\approx\) \(4.346738374\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.37587474575493, −12.03738906672128, −11.79736464370547, −11.25060995204742, −10.80193363571356, −10.41379628561221, −10.00655805746629, −9.198591441179182, −8.770117257854737, −8.424684106151167, −7.839436306560350, −7.386828432424986, −7.007279637027868, −6.508497166121561, −6.010967504546182, −5.369533105263636, −4.801536072368595, −4.658659515024879, −3.947086691725375, −3.560663181843830, −2.911404197524214, −2.549366315108460, −1.778309546106077, −1.069235602255086, −0.5279901899105087, 0.5279901899105087, 1.069235602255086, 1.778309546106077, 2.549366315108460, 2.911404197524214, 3.560663181843830, 3.947086691725375, 4.658659515024879, 4.801536072368595, 5.369533105263636, 6.010967504546182, 6.508497166121561, 7.007279637027868, 7.386828432424986, 7.839436306560350, 8.424684106151167, 8.770117257854737, 9.198591441179182, 10.00655805746629, 10.41379628561221, 10.80193363571356, 11.25060995204742, 11.79736464370547, 12.03738906672128, 12.37587474575493

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