Properties

Label 2-1850-1.1-c1-0-16
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 − 3.13·3-s + 4-s + 3.13·6-s − 4.13·7-s − 8-s + 6.81·9-s − 3.68·11-s − 3.13·12-s + 0.132·13-s + 4.13·14-s + 16-s + 5.13·17-s − 6.81·18-s + 0.451·19-s + 12.9·21-s + 3.68·22-s + 4.26·23-s + 3.13·24-s − 0.132·26-s − 11.9·27-s − 4.13·28-s − 10.2·29-s + 5.58·31-s − 32-s + 11.5·33-s − 5.13·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.80·3-s + 0.5·4-s + 1.27·6-s − 1.56·7-s − 0.353·8-s + 2.27·9-s − 1.10·11-s − 0.904·12-s + 0.0367·13-s + 1.10·14-s + 0.250·16-s + 1.24·17-s − 1.60·18-s + 0.103·19-s + 2.82·21-s + 0.784·22-s + 0.889·23-s + 0.639·24-s − 0.0260·26-s − 2.29·27-s − 0.780·28-s − 1.90·29-s + 1.00·31-s − 0.176·32-s + 2.00·33-s − 0.880·34-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 \( 1 + 3.13T + 3T^{2} \)
7 \( 1 + 4.13T + 7T^{2} \)
11 \( 1 + 3.68T + 11T^{2} \)
13 \( 1 - 0.132T + 13T^{2} \)
17 \( 1 - 5.13T + 17T^{2} \)
19 \( 1 - 0.451T + 19T^{2} \)
23 \( 1 - 4.26T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 5.58T + 31T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 5.07T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 4.39T + 61T^{2} \)
67 \( 1 - 0.228T + 67T^{2} \)
71 \( 1 - 9.49T + 71T^{2} \)
73 \( 1 + 3.54T + 73T^{2} \)
79 \( 1 + 0.903T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 - 0.777T + 89T^{2} \)
97 \( 1 + 0.638T + 97T^{2} \)
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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

−9.254897436761546222215162450805, −7.74601091024062655205762136804, −7.24700021762253524862626456436, −6.36033259953608977077414691603, −5.78132101808300318308149730057, −5.16424833681373195733162844713, −3.83630945070815497250331417604, −2.70795732868845213274132102179, −1.00378992318971350539312614864, 0, 1.00378992318971350539312614864, 2.70795732868845213274132102179, 3.83630945070815497250331417604, 5.16424833681373195733162844713, 5.78132101808300318308149730057, 6.36033259953608977077414691603, 7.24700021762253524862626456436, 7.74601091024062655205762136804, 9.254897436761546222215162450805

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