Properties

Label 2-1850-1.1-c1-0-35
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 3·11-s + 2·12-s + 4·13-s + 14-s + 16-s − 3·17-s + 18-s + 2·19-s + 2·21-s + 3·22-s − 6·23-s + 2·24-s + 4·26-s − 4·27-s + 28-s + 3·29-s + 5·31-s + 32-s + 6·33-s − 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s + 0.436·21-s + 0.639·22-s − 1.25·23-s + 0.408·24-s + 0.784·26-s − 0.769·27-s + 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 1.04·33-s − 0.514·34-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: yes
Arithmetic: yes
Character: $\chi_{1850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.432411577\)
\(L(\frac12)\) \(\approx\) \(4.432411577\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−9.179198904473865232269004873691, −8.239017656365015125223849506060, −7.985698124817949818204634866172, −6.67937370762302807973390678508, −6.19187304778673505127010809648, −5.01445637294381632948956822464, −4.03128883772338599220444916109, −3.47532993011179219314194223001, −2.43092253567530754938103868933, −1.45408956239590776662905952332, 1.45408956239590776662905952332, 2.43092253567530754938103868933, 3.47532993011179219314194223001, 4.03128883772338599220444916109, 5.01445637294381632948956822464, 6.19187304778673505127010809648, 6.67937370762302807973390678508, 7.985698124817949818204634866172, 8.239017656365015125223849506060, 9.179198904473865232269004873691

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