Properties

Label 2-1850-1.1-c1-0-34
Degree $2$
Conductor $1850$
Sign $1$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.62·3-s + 4-s + 2.62·6-s − 1.83·7-s + 8-s + 3.89·9-s + 4.19·11-s + 2.62·12-s − 0.369·13-s − 1.83·14-s + 16-s − 5.08·17-s + 3.89·18-s + 3.55·19-s − 4.81·21-s + 4.19·22-s + 5.62·23-s + 2.62·24-s − 0.369·26-s + 2.34·27-s − 1.83·28-s + 1.20·29-s + 10.1·31-s + 32-s + 11.0·33-s − 5.08·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.51·3-s + 0.5·4-s + 1.07·6-s − 0.692·7-s + 0.353·8-s + 1.29·9-s + 1.26·11-s + 0.757·12-s − 0.102·13-s − 0.489·14-s + 0.250·16-s − 1.23·17-s + 0.917·18-s + 0.816·19-s − 1.04·21-s + 0.893·22-s + 1.17·23-s + 0.535·24-s − 0.0724·26-s + 0.451·27-s − 0.346·28-s + 0.224·29-s + 1.81·31-s + 0.176·32-s + 1.91·33-s − 0.871·34-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: no
Arithmetic: yes
Character: $\chi_{1850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.602843583\)
\(L(\frac12)\) \(\approx\) \(4.602843583\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.62T + 3T^{2} \)
7 \( 1 + 1.83T + 7T^{2} \)
11 \( 1 - 4.19T + 11T^{2} \)
13 \( 1 + 0.369T + 13T^{2} \)
17 \( 1 + 5.08T + 17T^{2} \)
19 \( 1 - 3.55T + 19T^{2} \)
23 \( 1 - 5.62T + 23T^{2} \)
29 \( 1 - 1.20T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
41 \( 1 + 8.01T + 41T^{2} \)
43 \( 1 + 2.27T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 9.94T + 53T^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 + 9.79T + 61T^{2} \)
67 \( 1 - 1.85T + 67T^{2} \)
71 \( 1 - 2.86T + 71T^{2} \)
73 \( 1 + 8.09T + 73T^{2} \)
79 \( 1 - 6.06T + 79T^{2} \)
83 \( 1 + 8.93T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 6.05T + 97T^{2} \)
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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

−9.197722515630436105611758493532, −8.575952082750777824426924662200, −7.64417047746911667206271298516, −6.78098513328263379983282590623, −6.31233989340225536201941060685, −4.88412689989903802429346644503, −4.09363201413085186110958710718, −3.23518045657027919867410381213, −2.67121695803865695587107361899, −1.43679127594039923164431537595, 1.43679127594039923164431537595, 2.67121695803865695587107361899, 3.23518045657027919867410381213, 4.09363201413085186110958710718, 4.88412689989903802429346644503, 6.31233989340225536201941060685, 6.78098513328263379983282590623, 7.64417047746911667206271298516, 8.575952082750777824426924662200, 9.197722515630436105611758493532

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