Properties

Label 2-1850-1.1-c1-0-24
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 + 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: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.074469805\)
\(L(\frac12)\) \(\approx\) \(3.074469805\)
\(L(\frac{3}{2})\) 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

−9.085430276063378626463751307361, −8.512206978075076349174478737582, −7.61091526403152104785014718588, −6.70073776646501056286287484052, −5.95184061908039014224733690145, −5.23781168907357727327733294782, −4.22967748240561557188553702399, −3.43596770055792049299169169862, −2.44801223940381787647141426984, −1.16111311827070292245986629609, 1.16111311827070292245986629609, 2.44801223940381787647141426984, 3.43596770055792049299169169862, 4.22967748240561557188553702399, 5.23781168907357727327733294782, 5.95184061908039014224733690145, 6.70073776646501056286287484052, 7.61091526403152104785014718588, 8.512206978075076349174478737582, 9.085430276063378626463751307361

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