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Properties

Label 2-3200-40.13-c0-0-3
Degree $2$
Conductor $3200$
Sign $-0.130 - 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.130 - 0.991i$
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: \((2,\ 3200,\ (\ :0),\ -0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9730411300\)
\(L(\frac12)\) \(\approx\) \(0.9730411300\)
\(L(1)\) 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

−9.562102801596590507835922602005, −8.175576049819130900411158563058, −7.44848887122229869708016760380, −6.94662058611434887047048979921, −5.79161395380420576832955215056, −5.00677814025596232756310214299, −4.78244133679538291342670262986, −3.63111463087516152656123284271, −2.65340595301336041708800028876, −1.35239498874749541923251808711, 0.73649380079350580759001213942, 1.66107809232308796411211818102, 3.33916659617099566238518528290, 3.55606075210030279674597227654, 5.25153758421729062678156477675, 5.60736862900162336492999890315, 6.34152358577736916953797133627, 7.03697658223884712734253880223, 7.940707056313650635670843714392, 8.444536954889937821244948426029

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