Properties

Label 2-1850-1.1-c1-0-37
Degree $2$
Conductor $1850$
Sign $-1$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 3·9-s − 3·11-s + 4·13-s + 14-s + 16-s + 3·17-s + 3·18-s + 3·22-s + 8·23-s − 4·26-s − 28-s − 3·29-s − 7·31-s − 32-s − 3·34-s − 3·36-s + 37-s + 11·41-s − 11·43-s − 3·44-s − 8·46-s − 4·47-s − 6·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 0.904·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.707·18-s + 0.639·22-s + 1.66·23-s − 0.784·26-s − 0.188·28-s − 0.557·29-s − 1.25·31-s − 0.176·32-s − 0.514·34-s − 1/2·36-s + 0.164·37-s + 1.71·41-s − 1.67·43-s − 0.452·44-s − 1.17·46-s − 0.583·47-s − 6/7·49-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: yes
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 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−8.957628941716971024807124872679, −8.078853665107016721819448799239, −7.52235848830117702788047126274, −6.45089857042245298664114357857, −5.78161479488224941321715393153, −4.95605283514809047487763909166, −3.40811264738541382889276587488, −2.89285178790307178074659432324, −1.46859304329227136100061074818, 0, 1.46859304329227136100061074818, 2.89285178790307178074659432324, 3.40811264738541382889276587488, 4.95605283514809047487763909166, 5.78161479488224941321715393153, 6.45089857042245298664114357857, 7.52235848830117702788047126274, 8.078853665107016721819448799239, 8.957628941716971024807124872679

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