Properties

Degree $2$
Conductor $3150$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 4·11-s + 6·13-s + 14-s + 16-s + 2·17-s − 4·22-s + 6·26-s + 28-s − 6·29-s + 8·31-s + 32-s + 2·34-s + 10·37-s − 2·41-s − 4·43-s − 4·44-s + 8·47-s + 49-s + 6·52-s − 2·53-s + 56-s − 6·58-s + 8·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.852·22-s + 1.17·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s + 1/7·49-s + 0.832·52-s − 0.274·53-s + 0.133·56-s − 0.787·58-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3150} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.237547026\)
\(L(\frac12)\) \(\approx\) \(3.237547026\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−18.47112954489586, −18.14979440394476, −17.15430319001971, −16.57395236907679, −15.81701292749688, −15.45015983835004, −14.80139154882056, −13.92773527138079, −13.49998645793402, −12.94781318394889, −12.25129934248538, −11.34561056448748, −11.00821355083116, −10.26990662118516, −9.477312031842590, −8.363608030078191, −8.027088624732002, −7.150159014255240, −6.171502804116368, −5.665336182407942, −4.844008789854856, −3.990674749050463, −3.168044118194330, −2.227977856050313, −1.026993513510411, 1.026993513510411, 2.227977856050313, 3.168044118194330, 3.990674749050463, 4.844008789854856, 5.665336182407942, 6.171502804116368, 7.150159014255240, 8.027088624732002, 8.363608030078191, 9.477312031842590, 10.26990662118516, 11.00821355083116, 11.34561056448748, 12.25129934248538, 12.94781318394889, 13.49998645793402, 13.92773527138079, 14.80139154882056, 15.45015983835004, 15.81701292749688, 16.57395236907679, 17.15430319001971, 18.14979440394476, 18.47112954489586

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