Properties

Label 2-3200-8.5-c1-0-56
Degree $2$
Conductor $3200$
Sign $1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·3-s + 2.82·7-s − 2.00·9-s − 2.23i·11-s − 6.32i·13-s + 5·17-s + 2.23i·19-s + 6.32i·21-s − 5.65·23-s + 2.23i·27-s − 6.32i·29-s + 5.00·33-s − 6.32i·37-s + 14.1·39-s + 3·41-s + ⋯
L(s)  = 1  + 1.29i·3-s + 1.06·7-s − 0.666·9-s − 0.674i·11-s − 1.75i·13-s + 1.21·17-s + 0.512i·19-s + 1.38i·21-s − 1.17·23-s + 0.430i·27-s − 1.17i·29-s + 0.870·33-s − 1.03i·37-s + 2.26·39-s + 0.468·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.067602172\)
\(L(\frac12)\) \(\approx\) \(2.067602172\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT - 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 + 6.32iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 - 2.23iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 6.32iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6.32iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 8.94iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 12.6iT - 53T^{2} \)
59 \( 1 + 8.94iT - 59T^{2} \)
61 \( 1 - 6.32iT - 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 6.70iT - 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.528235648593454946915794748364, −8.062857124344856400936099975180, −7.51065414602691577555572046872, −5.99950637814402003534263259890, −5.49282252838308795576768562759, −4.90696250564629571546894774338, −3.83133781389755788404528064438, −3.43070761547423316832095447571, −2.15258750165526804654800071894, −0.68537427364798018976735953965, 1.29827131767529395022721206120, 1.73106325099570893526123908111, 2.72694223950730905013283491281, 4.15618811157245201920897066324, 4.75523619421423959895978080407, 5.79677986038035335978775471115, 6.60387982229154163232832969316, 7.23788856285094856755055337985, 7.78036994404254264475300009536, 8.439690379984450725001387532723

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