Properties

Label 2-1900-5.4-c1-0-5
Degree $2$
Conductor $1900$
Sign $0.447 - 0.894i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·7-s + 3·9-s − 4·11-s + 4i·13-s + 6i·17-s − 19-s + 2i·23-s + 6·29-s − 8·31-s + 4i·37-s + 6·41-s + 6i·43-s + 6i·47-s + 3·49-s − 8i·53-s + ⋯
L(s)  = 1  − 0.755i·7-s + 9-s − 1.20·11-s + 1.10i·13-s + 1.45i·17-s − 0.229·19-s + 0.417i·23-s + 1.11·29-s − 1.43·31-s + 0.657i·37-s + 0.937·41-s + 0.914i·43-s + 0.875i·47-s + 0.428·49-s − 1.09i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472989491\)
\(L(\frac12)\) \(\approx\) \(1.472989491\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−9.484368424030761845389022164638, −8.454940702545327898411804806210, −7.75649600970488012689430024740, −7.02282302209655139132229049351, −6.32249610706455623820169931528, −5.22957222326122652244461581748, −4.31132258830550929197656182777, −3.72864693903098481959129775750, −2.31587808798972415861147701726, −1.26536922468714984063166449458, 0.57436921608937453122972389544, 2.23650415450181533851702708922, 2.93964007791977598814731013201, 4.17357629543407708518549710199, 5.23966529840681822352569847175, 5.58196388526139325212439436531, 6.88673264989188868325553981914, 7.47898526201407817200021756921, 8.274552119508155899957419576267, 9.077293093414878039845065090700

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