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Properties

Label 2-3800-152.37-c0-0-11
Degree $2$
Conductor $3800$
Sign $0.923 - 0.382i$
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  + (0.923 + 0.382i)2-s − 0.765·3-s + (0.707 + 0.707i)4-s + (−0.707 − 0.292i)6-s + (0.382 + 0.923i)8-s − 0.414·9-s − 1.41i·11-s + (−0.541 − 0.541i)12-s + 1.84·13-s + i·16-s + (−0.382 − 0.158i)18-s − i·19-s + (0.541 − 1.30i)22-s + (−0.292 − 0.707i)24-s + (1.70 + 0.707i)26-s + 1.08·27-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s − 0.765·3-s + (0.707 + 0.707i)4-s + (−0.707 − 0.292i)6-s + (0.382 + 0.923i)8-s − 0.414·9-s − 1.41i·11-s + (−0.541 − 0.541i)12-s + 1.84·13-s + i·16-s + (−0.382 − 0.158i)18-s − i·19-s + (0.541 − 1.30i)22-s + (−0.292 − 0.707i)24-s + (1.70 + 0.707i)26-s + 1.08·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.851833756\)
\(L(\frac12)\) \(\approx\) \(1.851833756\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 + 0.765T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - 1.84T + T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.765T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 0.765T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 - 1.84T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.84iT - T^{2} \)
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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

−8.475708389463429544358323057732, −8.065435389225415936691970083255, −6.83577293307994652504774740958, −6.32336242781281004614150650287, −5.74740494317913117737195227010, −5.21899487176826105519053680157, −4.16808109241270629287230093756, −3.41459005076754594451332044936, −2.64547138984168035875865359867, −1.05240540401779602092582256845, 1.21349285112509968177632613427, 2.16452128086605089260542088091, 3.34796517526064993247471554112, 4.06303022128499761815118834342, 4.88255402410686897452378287105, 5.62159947276026298022484212294, 6.25094567046066021880446425599, 6.76710714686817530044982694043, 7.77728386759656982788556856705, 8.592349117201555057430490976925

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