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Properties

Label 2-3800-152.11-c0-0-2
Degree $2$
Conductor $3800$
Sign $-0.211 - 0.977i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + i·7-s + 0.999i·8-s + (0.5 + 0.866i)9-s − 11-s + (1.73 − i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + 1.99·26-s + (−0.866 + 0.499i)28-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + i·7-s + 0.999i·8-s + (0.5 + 0.866i)9-s − 11-s + (1.73 − i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + 1.99·26-s + (−0.866 + 0.499i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.211 - 0.977i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.211 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.237859185\)
\(L(\frac12)\) \(\approx\) \(2.237859185\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.428761861482699882193894562517, −8.154490171165269135437796560112, −7.46267607951210365773007206612, −6.47632951692374417876519499215, −5.73851315685485723796054475412, −5.28227786592011338161968098677, −4.52470547421764747266098938803, −3.41227592284891481411068724519, −2.76902908826341043465858410111, −1.76748742974104671036151173035, 1.01708103101501137684679887016, 1.90659575792252689265234237592, 3.23533613497745801465096609311, 3.91660813955462596166632400786, 4.32608151548375150930967587697, 5.45793327016459922882346401632, 6.23028220204478854160434697400, 6.74940506641070607224091097162, 7.58508436093900464672725234231, 8.447204416373063899717721441716

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