Properties

Label 2-3200-40.13-c0-0-0
Degree $2$
Conductor $3200$
Sign $-0.945 + 0.326i$
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  + (−0.707 + 0.707i)3-s − 1.73i·11-s + (−1.22 + 1.22i)17-s − 1.73·19-s + (−0.707 − 0.707i)27-s + (1.22 + 1.22i)33-s − 41-s + (−1.41 + 1.41i)43-s + i·49-s − 1.73i·51-s + (1.22 − 1.22i)57-s + (0.707 + 0.707i)67-s + (−1.22 − 1.22i)73-s + 1.00·81-s + (−0.707 + 0.707i)83-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 1.73i·11-s + (−1.22 + 1.22i)17-s − 1.73·19-s + (−0.707 − 0.707i)27-s + (1.22 + 1.22i)33-s − 41-s + (−1.41 + 1.41i)43-s + i·49-s − 1.73i·51-s + (1.22 − 1.22i)57-s + (0.707 + 0.707i)67-s + (−1.22 − 1.22i)73-s + 1.00·81-s + (−0.707 + 0.707i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.945 + 0.326i$
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: \((2,\ 3200,\ (\ :0),\ -0.945 + 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09795322717\)
\(L(\frac12)\) \(\approx\) \(0.09795322717\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.185155993327538183981208804966, −8.423298498817969566058894458310, −8.124465290040268167360235691519, −6.66660412777645552783803075721, −6.19405863365110866564627536922, −5.54394629790550381648404069422, −4.55539519236831630207771019810, −4.01984206917662877210301128932, −2.97537827091353287376745197482, −1.76151560694083765650570597611, 0.06028003099381638918542596440, 1.73728097273076743701399895757, 2.40006706866104849175705735381, 3.85717369936001509388542894062, 4.69591331680287043793723839086, 5.32769549576918200812543301685, 6.56034368080955807068010889324, 6.76619631979516292412399510908, 7.39059438730267053895841055745, 8.455983918559856241345702691080

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