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Properties

Label 2-3200-40.37-c0-0-0
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: $\chi_{3200} (1857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.537507318\)
\(L(\frac12)\) \(\approx\) \(1.537507318\)
\(L(1)\) 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

−9.065068559313306818621856336092, −8.317377467178684788760215940235, −7.72692586606656199104936462002, −6.75710675404685100932215814951, −6.08415644969605254369043268184, −4.95395884749949029801514043122, −4.22206333604225556196704049747, −3.67494598080229290651240763701, −2.56337491938000895277093103153, −1.65481608587261929640668256165, 0.880895053263373324823330517697, 2.16174029248050337525541524261, 2.96938910282853536737491927868, 3.71032120465903672724342247658, 4.90347729965299097238663303607, 5.75142274661073388479179566591, 6.46824260738211671990522566115, 7.37868696026063083408716807689, 7.928682473655379003471348593741, 8.680965434493629555124480800243

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