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Properties

Label 2-3200-8.3-c0-0-8
Degree $2$
Conductor $3200$
Sign $i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
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  − 9-s − 2i·13-s − 2i·37-s − 2·41-s + 49-s − 2i·53-s + 81-s + 2·89-s + 2i·117-s + ⋯
L(s)  = 1  − 9-s − 2i·13-s − 2i·37-s − 2·41-s + 49-s − 2i·53-s + 81-s + 2·89-s + 2i·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (2751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8984999906\)
\(L(\frac12)\) \(\approx\) \(0.8984999906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.536134325382482363974383738492, −8.030453763932426633951749954436, −7.27419047384194635585072462765, −6.29883582073430966244778405388, −5.49759100751325723434224819935, −5.11920698487912269616087548184, −3.74125111503461108477656430614, −3.10082029426782552552083614579, −2.15544367836647807850451860390, −0.53422476197666410419424320593, 1.52107575776039545426498858715, 2.53488290967964046997518051159, 3.50006639563732794187852070026, 4.44292960007245724644249300425, 5.14953829822516095702868730827, 6.18512722587648545371051526322, 6.66693385543387031370072944334, 7.52270870662828066904174145590, 8.486377504138129014191537040805, 8.918376648094086182329171603782

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