Properties

Label 2-1850-1.1-c1-0-42
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 + 1.53·3-s + 4-s + 1.53·6-s + 4.67·7-s + 8-s − 0.653·9-s + 0.0451·11-s + 1.53·12-s + 5.26·13-s + 4.67·14-s + 16-s + 3.60·17-s − 0.653·18-s − 6.22·19-s + 7.15·21-s + 0.0451·22-s − 2.20·23-s + 1.53·24-s + 5.26·26-s − 5.59·27-s + 4.67·28-s − 4.20·29-s − 3.01·31-s + 32-s + 0.0691·33-s + 3.60·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.884·3-s + 0.5·4-s + 0.625·6-s + 1.76·7-s + 0.353·8-s − 0.217·9-s + 0.0136·11-s + 0.442·12-s + 1.46·13-s + 1.24·14-s + 0.250·16-s + 0.875·17-s − 0.154·18-s − 1.42·19-s + 1.56·21-s + 0.00962·22-s − 0.459·23-s + 0.312·24-s + 1.03·26-s − 1.07·27-s + 0.882·28-s − 0.780·29-s − 0.540·31-s + 0.176·32-s + 0.0120·33-s + 0.618·34-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

\(L(1)\) \(\approx\) \(4.430170398\)
\(L(\frac12)\) \(\approx\) \(4.430170398\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.53T + 3T^{2} \)
7 \( 1 - 4.67T + 7T^{2} \)
11 \( 1 - 0.0451T + 11T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 - 3.60T + 17T^{2} \)
19 \( 1 + 6.22T + 19T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 + 4.20T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
41 \( 1 + 7.38T + 41T^{2} \)
43 \( 1 + 5.54T + 43T^{2} \)
47 \( 1 + 4.28T + 47T^{2} \)
53 \( 1 - 6.10T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 4.27T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 - 7.03T + 73T^{2} \)
79 \( 1 + 8.72T + 79T^{2} \)
83 \( 1 + 0.880T + 83T^{2} \)
89 \( 1 - 9.97T + 89T^{2} \)
97 \( 1 - 0.240T + 97T^{2} \)
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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

−8.904352806703044401284728677614, −8.262584487460450375114082737507, −8.004977542515655610760039337312, −6.88411339106705217998482809362, −5.82270825649933118868813829167, −5.18765671103623640760396080630, −4.09390755780666900007470533136, −3.53452258516760280778564800963, −2.24745682026215549323647861733, −1.52322316566165997344054467393, 1.52322316566165997344054467393, 2.24745682026215549323647861733, 3.53452258516760280778564800963, 4.09390755780666900007470533136, 5.18765671103623640760396080630, 5.82270825649933118868813829167, 6.88411339106705217998482809362, 8.004977542515655610760039337312, 8.262584487460450375114082737507, 8.904352806703044401284728677614

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