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Properties

Label 2-3200-8.5-c1-0-24
Degree $2$
Conductor $3200$
Sign $-0.707 - 0.707i$
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 − 2.00·9-s + 2.23i·11-s + 4i·13-s + 3·17-s − 2.23i·19-s + 8.94·23-s + 2.23i·27-s + 4i·29-s + 8.94·31-s − 5.00·33-s − 8i·37-s − 8.94·39-s − 5·41-s + 8.94i·43-s + ⋯
L(s)  = 1  + 1.29i·3-s − 0.666·9-s + 0.674i·11-s + 1.10i·13-s + 0.727·17-s − 0.512i·19-s + 1.86·23-s + 0.430i·27-s + 0.742i·29-s + 1.60·31-s − 0.870·33-s − 1.31i·37-s − 1.43·39-s − 0.780·41-s + 1.36i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.707 - 0.707i$
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: \((2,\ 3200,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.887696175\)
\(L(\frac12)\) \(\approx\) \(1.887696175\)
\(L(\frac{3}{2})\) 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 \( 1 + 7T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 2.23iT - 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 8.94iT - 43T^{2} \)
47 \( 1 - 8.94T + 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 8.94iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 - 6.70iT - 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 6.70iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.083189954188391603352110664339, −8.495308028949401692604129683660, −7.29004937044603921453560402002, −6.83773364390326744705495914983, −5.72834279761697451763529212966, −4.78286061471826263678987152399, −4.52271452759291246942313345114, −3.51115545853481729932681872616, −2.68425963713436699873147254644, −1.31011268342524137903603038099, 0.66403475056695630136571874300, 1.40466420492133063284002524212, 2.72442486495080174502595560066, 3.28912358528525055984153250031, 4.62288810544811788802928818007, 5.57926499046805634258681452068, 6.17531760093816033715539489449, 6.96747123870673604763334160411, 7.64267780013075829648274003253, 8.241085043155546754146743018127

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