Properties

Label 2-1400-5.4-c1-0-5
Degree $2$
Conductor $1400$
Sign $-0.894 - 0.447i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
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  + 2i·3-s + i·7-s − 9-s + 11-s + 4i·13-s − 6·19-s − 2·21-s + 3i·23-s + 4i·27-s + 3·29-s + 2i·33-s − 9i·37-s − 8·39-s + 2·41-s + 9i·43-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.377i·7-s − 0.333·9-s + 0.301·11-s + 1.10i·13-s − 1.37·19-s − 0.436·21-s + 0.625i·23-s + 0.769i·27-s + 0.557·29-s + 0.348i·33-s − 1.47i·37-s − 1.28·39-s + 0.312·41-s + 1.37i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.322561434\)
\(L(\frac12)\) \(\approx\) \(1.322561434\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 - 2iT - 3T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 9iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - iT - 67T^{2} \)
71 \( 1 + 7T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 9T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−9.697088823818481711701633072257, −9.272173387758320515469722329583, −8.595105853823862202606009325874, −7.51063017902087504211867564221, −6.50816979297899993781651137106, −5.73249939628901902563791392083, −4.52035633040959335322798926595, −4.22386622710022987780700289512, −3.03642885803833412542292928486, −1.74823676947384655770563592336, 0.53235295144991596471459908194, 1.73907920124756321652326457018, 2.82801833982336992243392563522, 4.03881938396490623702544436908, 5.07063496362356523054172266486, 6.28598407300574551655310136465, 6.66561719375102986601028406148, 7.65984502706950344450940204942, 8.204675737945985483454912992224, 9.017677189013623707133571075219

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