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Properties

Label 2-3200-40.13-c0-0-4
Degree $2$
Conductor $3200$
Sign $0.229 - 0.973i$
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  + (−1.41 + 1.41i)3-s − 3.00i·9-s + (2.82 + 2.82i)27-s + 2·41-s + (1.41 − 1.41i)43-s + i·49-s + (1.41 + 1.41i)67-s − 5.00·81-s + (−1.41 + 1.41i)83-s + 2i·89-s + (−1.41 − 1.41i)107-s + ⋯
L(s)  = 1  + (−1.41 + 1.41i)3-s − 3.00i·9-s + (2.82 + 2.82i)27-s + 2·41-s + (1.41 − 1.41i)43-s + i·49-s + (1.41 + 1.41i)67-s − 5.00·81-s + (−1.41 + 1.41i)83-s + 2i·89-s + (−1.41 − 1.41i)107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.229 - 0.973i$
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: \((2,\ 3200,\ (\ :0),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7171609862\)
\(L(\frac12)\) \(\approx\) \(0.7171609862\)
\(L(1)\) 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 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 2T + T^{2} \)
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 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
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

−9.345792579074569173699567312199, −8.494015596122020253857576034630, −7.30708584688644988817627490885, −6.53136495634829066566097288440, −5.72297536587405957385307798674, −5.32560041475320584520837822407, −4.27839128249732722841584331777, −3.94058827844325653207229184536, −2.74304594496652906932792203210, −0.902737017270382947585579771008, 0.74543994822053037423724224805, 1.80342857573806925923733378552, 2.73645256062189889466431933232, 4.28700261138015981597641656976, 5.08099743099059391598747660270, 5.89256752230771352010204573588, 6.32007069002753932815979250103, 7.19149568411239658681494494036, 7.67227275574834980072981229281, 8.394980667452418383204253627839

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