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Properties

Label 2-3200-40.13-c0-0-2
Degree $2$
Conductor $3200$
Sign $-0.525 - 0.850i$
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  + i·9-s + (−1 + i)13-s + (−1 + i)17-s − 2·29-s + (−1 − i)37-s + i·49-s + (1 − i)53-s + 2i·61-s + (1 + i)73-s − 81-s + (1 − i)97-s + (1 + i)113-s + (−1 − i)117-s + ⋯
L(s)  = 1  + i·9-s + (−1 + i)13-s + (−1 + i)17-s − 2·29-s + (−1 − i)37-s + i·49-s + (1 − i)53-s + 2i·61-s + (1 + i)73-s − 81-s + (1 − i)97-s + (1 + i)113-s + (−1 − i)117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7357164886\)
\(L(\frac12)\) \(\approx\) \(0.7357164886\)
\(L(1)\) 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 \( 1 - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.018526991209342829402860701965, −8.446464346721647461212176906445, −7.41660139795826743974563637905, −7.10170024991443914354257803399, −6.02860693883933144435531006341, −5.27287145102835076675358530759, −4.44390618092104692110982104190, −3.75663845108611906175238596697, −2.34613659879259446539720216092, −1.84982270941990949294265551047, 0.40864308719317885893019546828, 1.99630123309310964539548613852, 3.01384498110321250829265305199, 3.78070882516106446624104476266, 4.85389042418997605937052989961, 5.46263949197854850011078557801, 6.43461904558354509306971422290, 7.11457656648307350791939560277, 7.74922393383044929538967402887, 8.708610726679551382837449125380

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