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Properties

Label 2-3800-152.11-c0-0-1
Degree $2$
Conductor $3800$
Sign $0.977 - 0.211i$
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999·8-s + 2·11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (1 − 1.73i)22-s + (0.5 − 0.866i)24-s − 27-s + (0.499 + 0.866i)32-s + (−1 + 1.73i)33-s + (0.499 + 0.866i)34-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999·8-s + 2·11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (1 − 1.73i)22-s + (0.5 − 0.866i)24-s − 27-s + (0.499 + 0.866i)32-s + (−1 + 1.73i)33-s + (0.499 + 0.866i)34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.977 - 0.211i$
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: \((2,\ 3800,\ (\ :0),\ 0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367124572\)
\(L(\frac12)\) \(\approx\) \(1.367124572\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.946511456221940190918718945309, −8.302010359068744834789734413390, −6.91741247388986745227213042868, −6.23191229430000748422058641899, −5.58511415190801045012047297335, −4.67799771275447792930302113568, −3.89562819158692057959571383639, −3.75619954034887222552799991625, −2.20961884829812915988691692816, −1.31321151694277335264848694313, 0.799358572018411046409269381166, 2.14832972380656887458301148298, 3.43691899278919387479246264963, 4.21335207138998133787515816039, 4.94690648327987914644205900273, 6.06048054694742894635440594044, 6.38966044592526318461200754226, 7.11123658783965316685969147565, 7.43967993855636887693117970269, 8.703854960458849041661933009568

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