Properties

Label 2-3200-200.173-c0-0-1
Degree $2$
Conductor $3200$
Sign $-0.728 + 0.684i$
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  + (−0.951 + 0.309i)5-s + (0.587 − 0.809i)9-s + (−0.896 + 0.142i)13-s + (−1.76 − 0.896i)17-s + (0.809 − 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.309 − 1.95i)37-s + (−1.53 − 1.11i)41-s + (−0.309 + 0.951i)45-s + i·49-s + (−0.809 − 1.58i)53-s + (0.363 + 0.5i)61-s + (0.809 − 0.412i)65-s + (−0.896 − 0.142i)73-s + (−0.309 − 0.951i)81-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)5-s + (0.587 − 0.809i)9-s + (−0.896 + 0.142i)13-s + (−1.76 − 0.896i)17-s + (0.809 − 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.309 − 1.95i)37-s + (−1.53 − 1.11i)41-s + (−0.309 + 0.951i)45-s + i·49-s + (−0.809 − 1.58i)53-s + (0.363 + 0.5i)61-s + (0.809 − 0.412i)65-s + (−0.896 − 0.142i)73-s + (−0.309 − 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3879973948\)
\(L(\frac12)\) \(\approx\) \(0.3879973948\)
\(L(1)\) 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 \( 1 + (0.951 - 0.309i)T \)
good3 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)T^{2} \)
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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

−8.748392789636756702991710484177, −7.62528978160584760117439320474, −6.92651089877222477871138613692, −6.77418861180630627128800672426, −5.38515286828146472738462455871, −4.57468277406314491065732305745, −3.88716921322603659933419767258, −3.02729866320802390578541883153, −1.93773845971010464790778987993, −0.22394509308416144064359145954, 1.63400017221592345329292108300, 2.63336358978007568798857050090, 3.80498330943995031609068761867, 4.59945269496535077803201736256, 4.97226520101109751194688508896, 6.28311336000867066295811535383, 6.94972738588872957739191273066, 7.79660021561954848219919316235, 8.225156376007592145573192611396, 8.991085503392421638962172814115

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