Properties

Label 2-1850-1.1-c1-0-6
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 − 1.21·3-s + 4-s + 1.21·6-s + 3.59·7-s − 8-s − 1.52·9-s − 4.73·11-s − 1.21·12-s + 1.78·13-s − 3.59·14-s + 16-s + 1.83·17-s + 1.52·18-s + 3.28·19-s − 4.36·21-s + 4.73·22-s − 1.80·23-s + 1.21·24-s − 1.78·26-s + 5.49·27-s + 3.59·28-s − 0.755·29-s − 2.06·31-s − 32-s + 5.75·33-s − 1.83·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.701·3-s + 0.5·4-s + 0.495·6-s + 1.35·7-s − 0.353·8-s − 0.508·9-s − 1.42·11-s − 0.350·12-s + 0.495·13-s − 0.960·14-s + 0.250·16-s + 0.445·17-s + 0.359·18-s + 0.752·19-s − 0.951·21-s + 1.01·22-s − 0.376·23-s + 0.247·24-s − 0.350·26-s + 1.05·27-s + 0.678·28-s − 0.140·29-s − 0.371·31-s − 0.176·32-s + 1.00·33-s − 0.314·34-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

\(L(1)\) \(\approx\) \(0.9411390628\)
\(L(\frac12)\) \(\approx\) \(0.9411390628\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.21T + 3T^{2} \)
7 \( 1 - 3.59T + 7T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 - 1.78T + 13T^{2} \)
17 \( 1 - 1.83T + 17T^{2} \)
19 \( 1 - 3.28T + 19T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 + 2.06T + 31T^{2} \)
41 \( 1 + 2.57T + 41T^{2} \)
43 \( 1 - 9.19T + 43T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 - 6.56T + 53T^{2} \)
59 \( 1 - 9.19T + 59T^{2} \)
61 \( 1 + 9.39T + 61T^{2} \)
67 \( 1 + 2.39T + 67T^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 - 8.09T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 0.933T + 89T^{2} \)
97 \( 1 + 2.42T + 97T^{2} \)
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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

−9.151277243766888623845094382137, −8.307383448321807694381800379331, −7.85036462839381792939398339420, −7.09754322061188572025968622186, −5.77614325048529465289999736294, −5.48792846233368296067732696683, −4.53128558814086811483978676797, −3.10225356334277528534008017688, −2.02166940071156190352713018972, −0.75856436072018037282340273786, 0.75856436072018037282340273786, 2.02166940071156190352713018972, 3.10225356334277528534008017688, 4.53128558814086811483978676797, 5.48792846233368296067732696683, 5.77614325048529465289999736294, 7.09754322061188572025968622186, 7.85036462839381792939398339420, 8.307383448321807694381800379331, 9.151277243766888623845094382137

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