Properties

Label 2-1850-1.1-c1-0-44
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 + 2.73·3-s + 4-s + 2.73·6-s + 1.26·7-s + 8-s + 4.46·9-s + 1.46·11-s + 2.73·12-s − 1.46·13-s + 1.26·14-s + 16-s + 1.46·17-s + 4.46·18-s − 4.19·19-s + 3.46·21-s + 1.46·22-s + 8·23-s + 2.73·24-s − 1.46·26-s + 3.99·27-s + 1.26·28-s − 8.92·29-s − 2.73·31-s + 32-s + 4·33-s + 1.46·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.57·3-s + 0.5·4-s + 1.11·6-s + 0.479·7-s + 0.353·8-s + 1.48·9-s + 0.441·11-s + 0.788·12-s − 0.406·13-s + 0.338·14-s + 0.250·16-s + 0.355·17-s + 1.05·18-s − 0.962·19-s + 0.755·21-s + 0.312·22-s + 1.66·23-s + 0.557·24-s − 0.287·26-s + 0.769·27-s + 0.239·28-s − 1.65·29-s − 0.490·31-s + 0.176·32-s + 0.696·33-s + 0.251·34-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

\(L(1)\) \(\approx\) \(4.988401710\)
\(L(\frac12)\) \(\approx\) \(4.988401710\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 - 1.26T + 7T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 - 1.46T + 17T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + 8.92T + 29T^{2} \)
31 \( 1 + 2.73T + 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 6.92T + 43T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 0.196T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 - 5.26T + 79T^{2} \)
83 \( 1 - 5.26T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
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

−9.004001430377689944915927007538, −8.598733423047630184155344837903, −7.47499756616071603867196585268, −7.23218117195997713434059500337, −5.99270558720571648066210713444, −4.95707356291623409317770773447, −4.09544556474249645892883802569, −3.34847261523396353140969350432, −2.45575591700564970785754356198, −1.57118002568879529537407800860, 1.57118002568879529537407800860, 2.45575591700564970785754356198, 3.34847261523396353140969350432, 4.09544556474249645892883802569, 4.95707356291623409317770773447, 5.99270558720571648066210713444, 7.23218117195997713434059500337, 7.47499756616071603867196585268, 8.598733423047630184155344837903, 9.004001430377689944915927007538

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