Properties

Label 2-1850-1.1-c1-0-51
Degree $2$
Conductor $1850$
Sign $-1$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
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 − 2·7-s + 8-s − 3·9-s + 2·13-s − 2·14-s + 16-s − 6·17-s − 3·18-s − 6·19-s + 4·23-s + 2·26-s − 2·28-s − 4·31-s + 32-s − 6·34-s − 3·36-s − 37-s − 6·38-s − 10·41-s − 4·43-s + 4·46-s − 2·47-s − 3·49-s + 2·52-s + 2·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 9-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 1.37·19-s + 0.834·23-s + 0.392·26-s − 0.377·28-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s − 0.164·37-s − 0.973·38-s − 1.56·41-s − 0.609·43-s + 0.589·46-s − 0.291·47-s − 3/7·49-s + 0.277·52-s + 0.274·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
37 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−8.750324807682882988874725608634, −8.221535676991206294217792387889, −6.80613179515022113845924157121, −6.54855209691102148210096591792, −5.63131455176504702320472538865, −4.74788570792648033703597383465, −3.76886823851024934055265227029, −2.96349120039019999776178816015, −1.96778221189222956640787640137, 0, 1.96778221189222956640787640137, 2.96349120039019999776178816015, 3.76886823851024934055265227029, 4.74788570792648033703597383465, 5.63131455176504702320472538865, 6.54855209691102148210096591792, 6.80613179515022113845924157121, 8.221535676991206294217792387889, 8.750324807682882988874725608634

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