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Properties

Label 2-59850-1.1-c1-0-105
Degree $2$
Conductor $59850$
Sign $-1$
Analytic cond. $477.904$
Root an. cond. $21.8610$
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 + 4-s − 7-s − 8-s − 5·13-s + 14-s + 16-s + 6·17-s + 19-s + 6·23-s + 5·26-s − 28-s − 6·29-s + 2·31-s − 32-s − 6·34-s − 2·37-s − 38-s + 3·41-s − 8·43-s − 6·46-s + 3·47-s + 49-s − 5·52-s + 6·53-s + 56-s + 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 1.25·23-s + 0.980·26-s − 0.188·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.162·38-s + 0.468·41-s − 1.21·43-s − 0.884·46-s + 0.437·47-s + 1/7·49-s − 0.693·52-s + 0.824·53-s + 0.133·56-s + 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(477.904\)
Root analytic conductor: \(21.8610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 59850,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−14.76077988965778, −14.14435761194123, −13.54991901970620, −12.95874011673206, −12.37688959841713, −12.08165666964505, −11.55437732868813, −10.85929003603587, −10.42588131699043, −9.865305960042725, −9.429236445209016, −9.127971956469717, −8.290325215323075, −7.810762648001104, −7.294717868212298, −6.928621769942316, −6.253414817378743, −5.418017519194920, −5.241167532078903, −4.377173485148127, −3.529851967831799, −3.025837356867877, −2.433272625409344, −1.602984360688277, −0.8577772124200550, 0, 0.8577772124200550, 1.602984360688277, 2.433272625409344, 3.025837356867877, 3.529851967831799, 4.377173485148127, 5.241167532078903, 5.418017519194920, 6.253414817378743, 6.928621769942316, 7.294717868212298, 7.810762648001104, 8.290325215323075, 9.127971956469717, 9.429236445209016, 9.865305960042725, 10.42588131699043, 10.85929003603587, 11.55437732868813, 12.08165666964505, 12.37688959841713, 12.95874011673206, 13.54991901970620, 14.14435761194123, 14.76077988965778

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