Properties

Label 2-1850-1.1-c1-0-56
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.21·3-s + 4-s + 1.21·6-s − 3.59·7-s + 8-s − 1.52·9-s − 4.73·11-s + 1.21·12-s − 1.78·13-s − 3.59·14-s + 16-s − 1.83·17-s − 1.52·18-s + 3.28·19-s − 4.36·21-s − 4.73·22-s + 1.80·23-s + 1.21·24-s − 1.78·26-s − 5.49·27-s − 3.59·28-s − 0.755·29-s − 2.06·31-s + 32-s − 5.75·33-s − 1.83·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.701·3-s + 0.5·4-s + 0.495·6-s − 1.35·7-s + 0.353·8-s − 0.508·9-s − 1.42·11-s + 0.350·12-s − 0.495·13-s − 0.960·14-s + 0.250·16-s − 0.445·17-s − 0.359·18-s + 0.752·19-s − 0.951·21-s − 1.01·22-s + 0.376·23-s + 0.247·24-s − 0.350·26-s − 1.05·27-s − 0.678·28-s − 0.140·29-s − 0.371·31-s + 0.176·32-s − 1.00·33-s − 0.314·34-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: \(1\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.21T + 3T^{2} \)
7 \( 1 + 3.59T + 7T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 + 1.78T + 13T^{2} \)
17 \( 1 + 1.83T + 17T^{2} \)
19 \( 1 - 3.28T + 19T^{2} \)
23 \( 1 - 1.80T + 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 + 2.06T + 31T^{2} \)
41 \( 1 + 2.57T + 41T^{2} \)
43 \( 1 + 9.19T + 43T^{2} \)
47 \( 1 + 1.24T + 47T^{2} \)
53 \( 1 + 6.56T + 53T^{2} \)
59 \( 1 - 9.19T + 59T^{2} \)
61 \( 1 + 9.39T + 61T^{2} \)
67 \( 1 - 2.39T + 67T^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 + 8.09T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 0.933T + 89T^{2} \)
97 \( 1 - 2.42T + 97T^{2} \)
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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

−8.874687788959109329251705836694, −7.956457332172075125173072434867, −7.26658472144242846198002911503, −6.40785168792858709843699001390, −5.54105147213579645505073727464, −4.81663355257801189177141218422, −3.44971080724522410560992657218, −3.04175196976323395826458513540, −2.18772308432101248786939804207, 0, 2.18772308432101248786939804207, 3.04175196976323395826458513540, 3.44971080724522410560992657218, 4.81663355257801189177141218422, 5.54105147213579645505073727464, 6.40785168792858709843699001390, 7.26658472144242846198002911503, 7.956457332172075125173072434867, 8.874687788959109329251705836694

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