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Properties

Label 2-3800-152.37-c0-0-20
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

\(L(\frac{1}{2})\) \(\approx\) \(0.4475939614\)
\(L(\frac12)\) \(\approx\) \(0.4475939614\)
\(L(1)\) 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

−8.635646211470445520313143555663, −7.72992865645056296095989098670, −7.26392405593086597522988988069, −6.33095155164573700227583429306, −5.42222908942268707879390263925, −4.35318448124530525327542557942, −3.06514759957473651587086831663, −2.92772656027777916870413106862, −1.85773650239462392194805569996, −0.27668080574447311944771080826, 1.79224391649443448737962314976, 2.35281574647281016886865323434, 3.31356794907778796602483455846, 4.63950393980401036221728669385, 5.26770753772702138488774188281, 6.27868310914438135255482337081, 7.11017761750770467155617875529, 7.68309232837552690874360942997, 8.115206455270169776510059415688, 9.084558961400377828169450533688

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