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Properties

Label 2-3200-8.3-c0-0-10
Degree $2$
Conductor $3200$
Sign $0.707 + 0.707i$
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  + 1.41·3-s − 1.41i·7-s + 1.00·9-s − 2.00i·21-s − 1.41i·23-s + 1.41·43-s + 1.41i·47-s − 1.00·49-s + 2i·61-s − 1.41i·63-s + 1.41·67-s − 2.00i·69-s − 0.999·81-s − 1.41·83-s − 2·89-s + ⋯
L(s)  = 1  + 1.41·3-s − 1.41i·7-s + 1.00·9-s − 2.00i·21-s − 1.41i·23-s + 1.41·43-s + 1.41i·47-s − 1.00·49-s + 2i·61-s − 1.41i·63-s + 1.41·67-s − 2.00i·69-s − 0.999·81-s − 1.41·83-s − 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.707 + 0.707i$
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: \((2,\ 3200,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.002669591\)
\(L(\frac12)\) \(\approx\) \(2.002669591\)
\(L(1)\) 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 \( 1 - 1.41T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
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

−8.665737615479895382860413026625, −8.051883016671172821338307293323, −7.38227463155998729859186103476, −6.82949482905754132683824826191, −5.79234609971583775380940309516, −4.44276106109919712520248895877, −4.08906243896122972542059279983, −3.11462429774756731133955693690, −2.36145827616140930996402640706, −1.09712317869035619041231029461, 1.74723088761663398076278437162, 2.47553256079634359142274431985, 3.23308031101932703980607499082, 4.01190237004005075106141294896, 5.21766441807116065495789390897, 5.80504075805521325551252643694, 6.85349941111028680155885833770, 7.70523121478123607337963057388, 8.308700532174149077516814144806, 8.914343516922430647648003352701

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