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Properties

Label 2-3400-136.67-c0-0-4
Degree $2$
Conductor $3400$
Sign $0.5 - 0.866i$
Analytic cond. $1.69682$
Root an. cond. $1.30262$
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  − 2-s + 1.73i·3-s + 4-s − 1.73i·6-s − 8-s − 1.99·9-s − 1.73i·11-s + 1.73i·12-s + 16-s + (0.5 − 0.866i)17-s + 1.99·18-s + 19-s + 1.73i·22-s − 1.73i·24-s − 1.73i·27-s + ⋯
L(s)  = 1  − 2-s + 1.73i·3-s + 4-s − 1.73i·6-s − 8-s − 1.99·9-s − 1.73i·11-s + 1.73i·12-s + 16-s + (0.5 − 0.866i)17-s + 1.99·18-s + 19-s + 1.73i·22-s − 1.73i·24-s − 1.73i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (2651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ 0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8017078125\)
\(L(\frac12)\) \(\approx\) \(0.8017078125\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.056402805669477258596631684124, −8.404209732644910243364257872341, −7.78695609367521735742795523749, −6.68472479901453484304523322076, −5.67347435068572053191715688920, −5.36393957566869807667052840555, −4.13062940269282551602650599720, −3.21311437111088789647138916782, −2.79856551902569637085450036834, −0.873163239055474751422646823136, 0.986267311625149367007269361441, 1.90407178066539328671288275521, 2.45519607338942158229065846559, 3.69088918230921459155982029012, 5.21062489997373197573453862802, 6.03145623397622716056610672897, 6.74601685545353787565375654835, 7.48205750254631157204732195907, 7.59441119155831552696996950095, 8.500932636524943491894416705183

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