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Properties

Label 2-59850-1.1-c1-0-32
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 $0$

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 + 6·17-s + 19-s + 4·22-s + 4·23-s + 6·26-s − 28-s + 4·31-s − 32-s − 6·34-s + 6·37-s − 38-s + 6·41-s + 6·43-s − 4·44-s − 4·46-s + 8·47-s + 49-s − 6·52-s + 10·53-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 + 1.45·17-s + 0.229·19-s + 0.852·22-s + 0.834·23-s + 1.17·26-s − 0.188·28-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.986·37-s − 0.162·38-s + 0.937·41-s + 0.914·43-s − 0.603·44-s − 0.589·46-s + 1.16·47-s + 1/7·49-s − 0.832·52-s + 1.37·53-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: \(0\)
Selberg data: \((2,\ 59850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.447777852\)
\(L(\frac12)\) \(\approx\) \(1.447777852\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−14.36649317848382, −13.94979780986314, −13.11018326970868, −12.73766898643469, −12.26931668505381, −11.84181702502973, −11.16470834496074, −10.58353206456347, −10.12770785629484, −9.763578408096812, −9.297055932166663, −8.681741638143993, −7.862470434219209, −7.657979903932583, −7.233351871678942, −6.562110934582293, −5.759032002885096, −5.379696313717268, −4.814157713983147, −4.023644345939702, −3.171355929082512, −2.560851308386751, −2.336996483807855, −1.041774663169938, −0.5427553820190051, 0.5427553820190051, 1.041774663169938, 2.336996483807855, 2.560851308386751, 3.171355929082512, 4.023644345939702, 4.814157713983147, 5.379696313717268, 5.759032002885096, 6.562110934582293, 7.233351871678942, 7.657979903932583, 7.862470434219209, 8.681741638143993, 9.297055932166663, 9.763578408096812, 10.12770785629484, 10.58353206456347, 11.16470834496074, 11.84181702502973, 12.26931668505381, 12.73766898643469, 13.11018326970868, 13.94979780986314, 14.36649317848382

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