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Properties

Label 2-3200-40.19-c0-0-2
Degree $2$
Conductor $3200$
Sign $0.894 - 0.447i$
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  + 9-s + 2i·17-s + 2·41-s − 49-s − 2i·73-s + 81-s + 2·89-s + 2i·97-s + 2i·113-s + ⋯
L(s)  = 1  + 9-s + 2i·17-s + 2·41-s − 49-s − 2i·73-s + 81-s + 2·89-s + 2i·97-s + 2i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.355693605\)
\(L(\frac12)\) \(\approx\) \(1.355693605\)
\(L(1)\) 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 \( 1 - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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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

−8.963496949292052677763011377799, −7.999787932225622694317180560457, −7.58264317696518773638369638542, −6.51204956738730704559948272503, −6.07439773450342772959049841622, −4.99387653059748385391181737894, −4.16692752720597607692152074251, −3.53556836780382205191151687096, −2.22811695954816862519000804159, −1.30601088220893757435696801895, 0.966296917955607138933026412603, 2.25055935417778432185130372300, 3.16670819682663778767983452595, 4.24337099456036465231196824934, 4.85778357732164184091919133485, 5.72451586503508638787691703991, 6.71234694631139583741292751851, 7.29277222425626572370166565457, 7.889187846629726115227327439245, 8.926511194662820850240342560570

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