Properties

Label 2-3200-200.117-c0-0-1
Degree $2$
Conductor $3200$
Sign $-0.535 + 0.844i$
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  + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (−0.278 − 1.76i)17-s + (−0.309 − 0.951i)25-s + (0.5 − 0.363i)29-s + (−0.809 − 1.58i)37-s + (0.363 − 1.11i)41-s + (−0.809 + 0.587i)45-s i·49-s + (−0.309 − 0.0489i)53-s + (−1.53 + 0.5i)61-s + (−0.309 + 1.95i)65-s + (1.76 + 0.896i)73-s + (0.809 + 0.587i)81-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (−0.278 − 1.76i)17-s + (−0.309 − 0.951i)25-s + (0.5 − 0.363i)29-s + (−0.809 − 1.58i)37-s + (0.363 − 1.11i)41-s + (−0.809 + 0.587i)45-s i·49-s + (−0.309 − 0.0489i)53-s + (−1.53 + 0.5i)61-s + (−0.309 + 1.95i)65-s + (1.76 + 0.896i)73-s + (0.809 + 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8174715178\)
\(L(\frac12)\) \(\approx\) \(0.8174715178\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.278 + 1.76i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.587 + 0.809i)T^{2} \)
29 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \)
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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

−8.939332524590270077754794663965, −7.81677272919847641987148596867, −7.10609464206165661247332963511, −6.34702028093647513727798338385, −5.25895097093428642204177716159, −5.03134351098376661682499574530, −4.00589478731694201348028693452, −2.67611476656966156894670040029, −2.12193834580992001713722763947, −0.44826297439707773287003253848, 1.76960363162539352839939916410, 2.73289589462564306078027304816, 3.27105938648442573324135962377, 4.62718449959383427598071017231, 5.35604736326213731318133543229, 6.14916671452523482012374719017, 6.71274017716045485443569827646, 7.78508216438129598431138658010, 8.146938425186109621599093753143, 9.182229580049751600256388493750

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