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Properties

Label 2-127050-1.1-c1-0-89
Degree $2$
Conductor $127050$
Sign $-1$
Analytic cond. $1014.49$
Root an. cond. $31.8512$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 12-s − 2·13-s + 14-s + 16-s − 6·17-s − 18-s − 21-s + 8·23-s − 24-s + 2·26-s + 27-s − 28-s − 10·29-s − 8·31-s − 32-s + 6·34-s + 36-s − 2·37-s − 2·39-s + 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.218·21-s + 1.66·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.85·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1014.49\)
Root analytic conductor: \(31.8512\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 127050,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.64300161832469, −13.12376107838697, −12.86398116231271, −12.49120892985646, −11.61922164599286, −11.18762135912548, −10.89776252317985, −10.32853207083419, −9.679831366817203, −9.199252078836727, −9.026022479274549, −8.574816909614305, −7.760596085215337, −7.313520453133821, −7.096739401716673, −6.431299733772635, −5.849111227056232, −5.190978966086558, −4.621392759407529, −3.945936422469202, −3.331515251802374, −2.833583962000726, −2.065992482376629, −1.784216981850386, −0.7376539198798222, 0, 0.7376539198798222, 1.784216981850386, 2.065992482376629, 2.833583962000726, 3.331515251802374, 3.945936422469202, 4.621392759407529, 5.190978966086558, 5.849111227056232, 6.431299733772635, 7.096739401716673, 7.313520453133821, 7.760596085215337, 8.574816909614305, 9.026022479274549, 9.199252078836727, 9.679831366817203, 10.32853207083419, 10.89776252317985, 11.18762135912548, 11.61922164599286, 12.49120892985646, 12.86398116231271, 13.12376107838697, 13.64300161832469

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