Properties

Label 2-1850-1.1-c1-0-19
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 − 0.332·3-s + 4-s − 0.332·6-s + 3.51·7-s + 8-s − 2.88·9-s − 0.290·11-s − 0.332·12-s − 7.12·13-s + 3.51·14-s + 16-s + 6.17·17-s − 2.88·18-s + 5.83·19-s − 1.16·21-s − 0.290·22-s + 6.45·23-s − 0.332·24-s − 7.12·26-s + 1.96·27-s + 3.51·28-s − 1.18·29-s + 9.77·31-s + 32-s + 0.0965·33-s + 6.17·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.192·3-s + 0.5·4-s − 0.135·6-s + 1.32·7-s + 0.353·8-s − 0.963·9-s − 0.0874·11-s − 0.0961·12-s − 1.97·13-s + 0.938·14-s + 0.250·16-s + 1.49·17-s − 0.680·18-s + 1.33·19-s − 0.255·21-s − 0.0618·22-s + 1.34·23-s − 0.0679·24-s − 1.39·26-s + 0.377·27-s + 0.663·28-s − 0.219·29-s + 1.75·31-s + 0.176·32-s + 0.0168·33-s + 1.05·34-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

\(L(1)\) \(\approx\) \(2.816481454\)
\(L(\frac12)\) \(\approx\) \(2.816481454\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.332T + 3T^{2} \)
7 \( 1 - 3.51T + 7T^{2} \)
11 \( 1 + 0.290T + 11T^{2} \)
13 \( 1 + 7.12T + 13T^{2} \)
17 \( 1 - 6.17T + 17T^{2} \)
19 \( 1 - 5.83T + 19T^{2} \)
23 \( 1 - 6.45T + 23T^{2} \)
29 \( 1 + 1.18T + 29T^{2} \)
31 \( 1 - 9.77T + 31T^{2} \)
41 \( 1 - 1.64T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 + 5.69T + 47T^{2} \)
53 \( 1 + 9.32T + 53T^{2} \)
59 \( 1 - 5.94T + 59T^{2} \)
61 \( 1 + 1.27T + 61T^{2} \)
67 \( 1 + 6.04T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 + 1.71T + 73T^{2} \)
79 \( 1 - 8.25T + 79T^{2} \)
83 \( 1 + 2.41T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−9.370541805546412443596429245393, −8.106833002415794849816185838713, −7.72808455824006689457734665674, −6.90581022761955770365413685303, −5.68128685559181352174083761097, −5.10086051785199634399489963118, −4.68162318107069891999525266141, −3.19347351766014019075177752912, −2.52313881840677482177598477001, −1.10256475422149696906701825949, 1.10256475422149696906701825949, 2.52313881840677482177598477001, 3.19347351766014019075177752912, 4.68162318107069891999525266141, 5.10086051785199634399489963118, 5.68128685559181352174083761097, 6.90581022761955770365413685303, 7.72808455824006689457734665674, 8.106833002415794849816185838713, 9.370541805546412443596429245393

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