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Properties

Label 2-3800-760.189-c0-0-0
Degree $2$
Conductor $3800$
Sign $0.447 + 0.894i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
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  + i·2-s + 1.87i·3-s − 4-s − 1.87·6-s + 1.53i·7-s − i·8-s − 2.53·9-s − 1.87i·12-s − 0.347i·13-s − 1.53·14-s + 16-s + 0.347i·17-s − 2.53i·18-s − 19-s − 2.87·21-s + ⋯
L(s)  = 1  + i·2-s + 1.87i·3-s − 4-s − 1.87·6-s + 1.53i·7-s − i·8-s − 2.53·9-s − 1.87i·12-s − 0.347i·13-s − 1.53·14-s + 16-s + 0.347i·17-s − 2.53i·18-s − 19-s − 2.87·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7156713648\)
\(L(\frac12)\) \(\approx\) \(0.7156713648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 1.87iT - T^{2} \)
7 \( 1 - 1.53iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 0.347iT - T^{2} \)
17 \( 1 - 0.347iT - T^{2} \)
23 \( 1 - 1.87iT - T^{2} \)
29 \( 1 + 0.347T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + 1.53iT - T^{2} \)
59 \( 1 - 1.87T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.53iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.87iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.221520604104487920005824231461, −8.543981947821450636245373687629, −8.245347094212578758597246758267, −6.99228045358536013562765072263, −5.84698456209305793212209979752, −5.56950354908965130297355905090, −5.02982893561396890453227571808, −3.99365414189022273243799070410, −3.50683843871902413610775112285, −2.35101914896618355435565505983, 0.42637819116909785239484517734, 1.31404186641320955428549139283, 2.21520706055977960957688491061, 3.02063651519615724527946498287, 4.11195832858536234956969784778, 4.84987607834863039972593962780, 6.11115827613248193099251375563, 6.67104757455432379022198614489, 7.39930596895558060370200824587, 8.073465492646310916259054136405

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