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 − 2·13-s − 14-s + 16-s + 2·19-s − 2·26-s − 28-s + 6·29-s + 8·31-s + 32-s + 4·37-s + 2·38-s + 6·41-s − 2·43-s + 6·47-s + 49-s − 2·52-s − 6·53-s − 56-s + 6·58-s + 12·59-s + 8·61-s + 8·62-s + 64-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 + 0.458·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.657·37-s + 0.324·38-s + 0.937·41-s − 0.304·43-s + 0.875·47-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 + 1.02·61-s + 1.01·62-s + 1/8·64-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

\(L(1)\) \(\approx\) \(2.854621188\)
\(L(\frac12)\) \(\approx\) \(2.854621188\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−18.69782962107444, −17.75717226738777, −17.30617827135202, −16.54420596911288, −15.80794991902050, −15.56640983809801, −14.49203987230320, −14.28471312590955, −13.35987578283546, −12.94125436855191, −12.06840383171050, −11.74921661506400, −10.84174574664036, −10.09455231471896, −9.575678415871646, −8.583295435803265, −7.834847884920724, −7.032431907056678, −6.381852953525569, −5.606553087385689, −4.773659690383117, −4.073814214807390, −3.036570738714733, −2.377137567171889, −0.9326122112635866, 0.9326122112635866, 2.377137567171889, 3.036570738714733, 4.073814214807390, 4.773659690383117, 5.606553087385689, 6.381852953525569, 7.032431907056678, 7.834847884920724, 8.583295435803265, 9.575678415871646, 10.09455231471896, 10.84174574664036, 11.74921661506400, 12.06840383171050, 12.94125436855191, 13.35987578283546, 14.28471312590955, 14.49203987230320, 15.56640983809801, 15.80794991902050, 16.54420596911288, 17.30617827135202, 17.75717226738777, 18.69782962107444

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