Properties

Label 2-1850-1.1-c1-0-36
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 − 0.140·3-s + 4-s + 0.140·6-s − 1.14·7-s − 8-s − 2.98·9-s + 3.12·11-s − 0.140·12-s − 2.85·13-s + 1.14·14-s + 16-s + 2.14·17-s + 2.98·18-s + 4.26·19-s + 0.160·21-s − 3.12·22-s − 1.71·23-s + 0.140·24-s + 2.85·26-s + 0.839·27-s − 1.14·28-s − 4.28·29-s + 6.40·31-s − 32-s − 0.438·33-s − 2.14·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0810·3-s + 0.5·4-s + 0.0573·6-s − 0.431·7-s − 0.353·8-s − 0.993·9-s + 0.940·11-s − 0.0405·12-s − 0.793·13-s + 0.304·14-s + 0.250·16-s + 0.519·17-s + 0.702·18-s + 0.977·19-s + 0.0349·21-s − 0.665·22-s − 0.358·23-s + 0.0286·24-s + 0.560·26-s + 0.161·27-s − 0.215·28-s − 0.794·29-s + 1.14·31-s − 0.176·32-s − 0.0762·33-s − 0.367·34-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 \( 1 + 0.140T + 3T^{2} \)
7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 + 2.85T + 13T^{2} \)
17 \( 1 - 2.14T + 17T^{2} \)
19 \( 1 - 4.26T + 19T^{2} \)
23 \( 1 + 1.71T + 23T^{2} \)
29 \( 1 + 4.28T + 29T^{2} \)
31 \( 1 - 6.40T + 31T^{2} \)
41 \( 1 + 3.24T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 7.96T + 53T^{2} \)
59 \( 1 - 2.69T + 59T^{2} \)
61 \( 1 - 4.57T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + 7.10T + 71T^{2} \)
73 \( 1 - 0.261T + 73T^{2} \)
79 \( 1 + 8.52T + 79T^{2} \)
83 \( 1 - 1.16T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−8.876594542691850911164641915233, −8.173182435395257165521158046602, −7.34180286149514231766113099271, −6.54505990865091987913140605835, −5.80446653765215366337044379923, −4.88423769562506654079926811634, −3.53808184843350127664256815147, −2.79500626875052348091352224436, −1.46839150036347553571859932344, 0, 1.46839150036347553571859932344, 2.79500626875052348091352224436, 3.53808184843350127664256815147, 4.88423769562506654079926811634, 5.80446653765215366337044379923, 6.54505990865091987913140605835, 7.34180286149514231766113099271, 8.173182435395257165521158046602, 8.876594542691850911164641915233

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