Properties

Label 2-1850-1.1-c1-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.53·3-s + 4-s − 1.53·6-s + 2.87·7-s + 8-s − 0.630·9-s − 1.09·11-s − 1.53·12-s − 4.53·13-s + 2.87·14-s + 16-s − 2.80·17-s − 0.630·18-s − 5.04·19-s − 4.43·21-s − 1.09·22-s − 7.41·23-s − 1.53·24-s − 4.53·26-s + 5.58·27-s + 2.87·28-s + 6.68·29-s + 3.51·31-s + 32-s + 1.68·33-s − 2.80·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.888·3-s + 0.5·4-s − 0.628·6-s + 1.08·7-s + 0.353·8-s − 0.210·9-s − 0.329·11-s − 0.444·12-s − 1.25·13-s + 0.769·14-s + 0.250·16-s − 0.679·17-s − 0.148·18-s − 1.15·19-s − 0.967·21-s − 0.232·22-s − 1.54·23-s − 0.314·24-s − 0.890·26-s + 1.07·27-s + 0.544·28-s + 1.24·29-s + 0.630·31-s + 0.176·32-s + 0.292·33-s − 0.480·34-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: \(1\)
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 \( 1 + 1.53T + 3T^{2} \)
7 \( 1 - 2.87T + 7T^{2} \)
11 \( 1 + 1.09T + 11T^{2} \)
13 \( 1 + 4.53T + 13T^{2} \)
17 \( 1 + 2.80T + 17T^{2} \)
19 \( 1 + 5.04T + 19T^{2} \)
23 \( 1 + 7.41T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 3.51T + 31T^{2} \)
41 \( 1 + 8.07T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 6.29T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 - 6.29T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 2.58T + 79T^{2} \)
83 \( 1 + 8.48T + 83T^{2} \)
89 \( 1 + 6.51T + 89T^{2} \)
97 \( 1 + 3.07T + 97T^{2} \)
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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

−8.494031857496862585393605755609, −8.166922527811166607370359984698, −6.96556838448942531042278721785, −6.38647032776418567834371524018, −5.41035212151330347280175613963, −4.84772763250043229420889753281, −4.22090296842033137633880042530, −2.73270499268890823377421484147, −1.83267338175141695976612589082, 0, 1.83267338175141695976612589082, 2.73270499268890823377421484147, 4.22090296842033137633880042530, 4.84772763250043229420889753281, 5.41035212151330347280175613963, 6.38647032776418567834371524018, 6.96556838448942531042278721785, 8.166922527811166607370359984698, 8.494031857496862585393605755609

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