Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s − 2·26-s − 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s − 2·37-s − 4·38-s − 6·41-s − 8·43-s − 12·47-s + 49-s − 2·52-s + 6·53-s − 56-s + 6·58-s + 12·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.277·52-s + 0.824·53-s − 0.133·56-s + 0.787·58-s + 1.56·59-s + 0.256·61-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: \(1\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -1)\)

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

−19.10191777534854, −18.07184693158180, −17.66688706812109, −16.79654499219246, −16.34416920040033, −15.55455869110539, −15.07323821372112, −14.48183596511669, −13.70888097661350, −13.03433170144647, −12.74791589908781, −11.71817186549171, −11.40476399462309, −10.33909737785192, −10.03830065066432, −8.845843570702888, −8.441852246728578, −7.278593507465635, −6.716380413533202, −6.114437678286766, −5.064861440359341, −4.495115166959840, −3.591307154225099, −2.653221769914648, −1.793002210675610, 0, 1.793002210675610, 2.653221769914648, 3.591307154225099, 4.495115166959840, 5.064861440359341, 6.114437678286766, 6.716380413533202, 7.278593507465635, 8.441852246728578, 8.845843570702888, 10.03830065066432, 10.33909737785192, 11.40476399462309, 11.71817186549171, 12.74791589908781, 13.03433170144647, 13.70888097661350, 14.48183596511669, 15.07323821372112, 15.55455869110539, 16.34416920040033, 16.79654499219246, 17.66688706812109, 18.07184693158180, 19.10191777534854

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