Properties

Label 2-1400-5.4-c1-0-14
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  i·7-s + 3·9-s + 11-s − 2i·13-s + 4i·17-s + 2·19-s − 5i·23-s − 29-s − 2·31-s + 3i·37-s + 12·41-s − 11i·43-s + 2i·47-s − 49-s − 6i·53-s + ⋯
L(s)  = 1  − 0.377i·7-s + 9-s + 0.301·11-s − 0.554i·13-s + 0.970i·17-s + 0.458·19-s − 1.04i·23-s − 0.185·29-s − 0.359·31-s + 0.493i·37-s + 1.87·41-s − 1.67i·43-s + 0.291i·47-s − 0.142·49-s − 0.824i·53-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

\(L(1)\) \(\approx\) \(1.848585316\)
\(L(\frac12)\) \(\approx\) \(1.848585316\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 11iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 9T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.599061180425758809174744402252, −8.678923886928553490258809363353, −7.84225347065163496641354453421, −7.09184273427207850064323285530, −6.30961418212361023393804338565, −5.31246900451201230939971897737, −4.27705929900894322708507356213, −3.59575598684042770727745005932, −2.19880435768382827313471797407, −0.929880260616596768860034320528, 1.19435034322577964247896542003, 2.40480825458192401725560936454, 3.63389025602634098694874082133, 4.52952872578545709568449537331, 5.43204864951465508273921149027, 6.39925925855325424010351243521, 7.28583188790213835075936085875, 7.81223633686724022463121525964, 9.206859942768083291343977409771, 9.358228634353551045451433032501

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