Properties

Label 2-1850-1.1-c1-0-55
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 + 2.27·3-s + 4-s − 2.27·6-s + 1.27·7-s − 8-s + 2.16·9-s − 4.43·11-s + 2.27·12-s − 5.27·13-s − 1.27·14-s + 16-s − 0.273·17-s − 2.16·18-s − 5.71·19-s + 2.89·21-s + 4.43·22-s − 6.54·23-s − 2.27·24-s + 5.27·26-s − 1.89·27-s + 1.27·28-s + 0.546·29-s − 5.98·31-s − 32-s − 10.0·33-s + 0.273·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.31·3-s + 0.5·4-s − 0.927·6-s + 0.481·7-s − 0.353·8-s + 0.722·9-s − 1.33·11-s + 0.656·12-s − 1.46·13-s − 0.340·14-s + 0.250·16-s − 0.0662·17-s − 0.510·18-s − 1.31·19-s + 0.631·21-s + 0.946·22-s − 1.36·23-s − 0.463·24-s + 1.03·26-s − 0.364·27-s + 0.240·28-s + 0.101·29-s − 1.07·31-s − 0.176·32-s − 1.75·33-s + 0.0468·34-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 \( 1 - 2.27T + 3T^{2} \)
7 \( 1 - 1.27T + 7T^{2} \)
11 \( 1 + 4.43T + 11T^{2} \)
13 \( 1 + 5.27T + 13T^{2} \)
17 \( 1 + 0.273T + 17T^{2} \)
19 \( 1 + 5.71T + 19T^{2} \)
23 \( 1 + 6.54T + 23T^{2} \)
29 \( 1 - 0.546T + 29T^{2} \)
31 \( 1 + 5.98T + 31T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2.33T + 53T^{2} \)
59 \( 1 - 2.37T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 7.15T + 67T^{2} \)
71 \( 1 - 5.60T + 71T^{2} \)
73 \( 1 + 9.71T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 3.89T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 0.879T + 97T^{2} \)
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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

−8.734671166136427204870632918819, −7.994495575888687250059781649198, −7.77248253254193019822401590837, −6.83465208904565015238105270550, −5.62578311779360009185543949590, −4.66867185435946907554061808349, −3.56668292369494028811310697636, −2.31772836902885995639686810395, −2.17219856844586864990858923792, 0, 2.17219856844586864990858923792, 2.31772836902885995639686810395, 3.56668292369494028811310697636, 4.66867185435946907554061808349, 5.62578311779360009185543949590, 6.83465208904565015238105270550, 7.77248253254193019822401590837, 7.994495575888687250059781649198, 8.734671166136427204870632918819

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