Properties

Label 2-1900-380.303-c0-0-3
Degree $2$
Conductor $1900$
Sign $0.229 - 0.973i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 − 0.382i)2-s + (1.30 + 1.30i)3-s + (0.707 + 0.707i)4-s + (−0.707 − 1.70i)6-s + (−0.382 − 0.923i)8-s + 2.41i·9-s + 1.41i·11-s + 1.84i·12-s + (1.30 − 1.30i)13-s + i·16-s + (0.923 − 2.23i)18-s − 19-s + (0.541 − 1.30i)22-s + (0.707 − 1.70i)24-s + (−1.70 + 0.707i)26-s + (−1.84 + 1.84i)27-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (1.30 + 1.30i)3-s + (0.707 + 0.707i)4-s + (−0.707 − 1.70i)6-s + (−0.382 − 0.923i)8-s + 2.41i·9-s + 1.41i·11-s + 1.84i·12-s + (1.30 − 1.30i)13-s + i·16-s + (0.923 − 2.23i)18-s − 19-s + (0.541 − 1.30i)22-s + (0.707 − 1.70i)24-s + (−1.70 + 0.707i)26-s + (−1.84 + 1.84i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.202410270\)
\(L(\frac12)\) \(\approx\) \(1.202410270\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
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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

−9.549290647438705855246011974183, −8.856693998678244023882765798082, −8.271814660746980113071034843154, −7.72160249400714515255961300478, −6.70204724527445656439742935505, −5.38059162060606303561619986827, −4.23783040191009960272644619732, −3.62876461684270382202729981966, −2.72472027568448686381534945357, −1.83049524918377246948289925169, 1.11433052617015332319862087922, 1.96924812133161843190386158484, 2.98553796850816700673536892321, 3.95121348399835033493902104765, 5.75527321953177669088138829452, 6.56325545125810467420952774761, 6.82512687009536763772452383368, 8.024690108818522441651771848254, 8.381279074766846867721174095129, 8.908090324624514864140297186093

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