Properties

Label 2-1850-1.1-c1-0-21
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 − 2.44·3-s + 4-s − 2.44·6-s + 4.44·7-s + 8-s + 2.99·9-s + 4.89·11-s − 2.44·12-s + 4·13-s + 4.44·14-s + 16-s − 4.89·17-s + 2.99·18-s + 3.55·19-s − 10.8·21-s + 4.89·22-s − 8.89·23-s − 2.44·24-s + 4·26-s + 4.44·28-s − 1.55·31-s + 32-s − 11.9·33-s − 4.89·34-s + 2.99·36-s + 37-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.999·6-s + 1.68·7-s + 0.353·8-s + 0.999·9-s + 1.47·11-s − 0.707·12-s + 1.10·13-s + 1.18·14-s + 0.250·16-s − 1.18·17-s + 0.707·18-s + 0.814·19-s − 2.37·21-s + 1.04·22-s − 1.85·23-s − 0.499·24-s + 0.784·26-s + 0.840·28-s − 0.278·31-s + 0.176·32-s − 2.08·33-s − 0.840·34-s + 0.499·36-s + 0.164·37-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

\(L(1)\) \(\approx\) \(2.308313673\)
\(L(\frac12)\) \(\approx\) \(2.308313673\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.44T + 3T^{2} \)
7 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 3.55T + 19T^{2} \)
23 \( 1 + 8.89T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 1.55T + 31T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 4.44T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 5.55T + 67T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 6.44T + 79T^{2} \)
83 \( 1 + 9.55T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.235291918252385620770789855389, −8.351904345703161987307602902443, −7.48367395725392788651788867020, −6.45698737624112050522683710410, −6.03523087644994316540443736665, −5.19176673177871348612719735805, −4.40385530484308334981962436257, −3.84452367022871124691960758388, −1.97516987234776102347959159132, −1.10859637255207741574631887087, 1.10859637255207741574631887087, 1.97516987234776102347959159132, 3.84452367022871124691960758388, 4.40385530484308334981962436257, 5.19176673177871348612719735805, 6.03523087644994316540443736665, 6.45698737624112050522683710410, 7.48367395725392788651788867020, 8.351904345703161987307602902443, 9.235291918252385620770789855389

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