Properties

Label 2-1850-1.1-c1-0-43
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 + 3.44·3-s + 4-s + 3.44·6-s − 2.44·7-s + 8-s + 8.89·9-s − 1.44·11-s + 3.44·12-s + 4.44·13-s − 2.44·14-s + 16-s − 1.44·17-s + 8.89·18-s − 5·19-s − 8.44·21-s − 1.44·22-s + 2·23-s + 3.44·24-s + 4.44·26-s + 20.3·27-s − 2.44·28-s + 8.89·29-s − 0.449·31-s + 32-s − 5·33-s − 1.44·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.99·3-s + 0.5·4-s + 1.40·6-s − 0.925·7-s + 0.353·8-s + 2.96·9-s − 0.437·11-s + 0.995·12-s + 1.23·13-s − 0.654·14-s + 0.250·16-s − 0.351·17-s + 2.09·18-s − 1.14·19-s − 1.84·21-s − 0.309·22-s + 0.417·23-s + 0.704·24-s + 0.872·26-s + 3.91·27-s − 0.462·28-s + 1.65·29-s − 0.0807·31-s + 0.176·32-s − 0.870·33-s − 0.248·34-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

\(L(1)\) \(\approx\) \(5.227273380\)
\(L(\frac12)\) \(\approx\) \(5.227273380\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.44T + 3T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 + 1.44T + 11T^{2} \)
13 \( 1 - 4.44T + 13T^{2} \)
17 \( 1 + 1.44T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 8.89T + 29T^{2} \)
31 \( 1 + 0.449T + 31T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 + 1.55T + 61T^{2} \)
67 \( 1 - 9.44T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 6.79T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 1.44T + 83T^{2} \)
89 \( 1 + 0.348T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.942923879399788595071668107048, −8.561939574147500571790315438538, −7.80974681604262524886903360551, −6.75553309758946923431375030008, −6.38428722623152595777496316589, −4.82604220680341334907637978048, −3.99537293944878661435748904089, −3.25827034717114569518100019075, −2.65893092684107510662709321005, −1.56276024939701118332757226885, 1.56276024939701118332757226885, 2.65893092684107510662709321005, 3.25827034717114569518100019075, 3.99537293944878661435748904089, 4.82604220680341334907637978048, 6.38428722623152595777496316589, 6.75553309758946923431375030008, 7.80974681604262524886903360551, 8.561939574147500571790315438538, 8.942923879399788595071668107048

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