↑

Properties

Label 2-1400-7.4-c1-0-3
Degree $2$
Conductor $1400$
Sign $-0.0725 - 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 − 0.358i)3-s + (−1.62 − 2.09i)7-s + (1.41 + 2.44i)9-s + (−0.414 + 0.717i)11-s − 2·13-s + (−3.82 + 6.63i)17-s + (2.82 + 4.89i)19-s + (−1.08 + 0.148i)21-s + (−2.79 − 4.83i)23-s + 2.41·27-s − 7.82·29-s + (−0.414 + 0.717i)31-s + (0.171 + 0.297i)33-s + (2.82 + 4.89i)37-s + (−0.414 + 0.717i)39-s + ⋯
L(s)  = 1  + (0.119 − 0.207i)3-s + (−0.612 − 0.790i)7-s + (0.471 + 0.816i)9-s + (−0.124 + 0.216i)11-s − 0.554·13-s + (−0.928 + 1.60i)17-s + (0.648 + 1.12i)19-s + (−0.236 + 0.0324i)21-s + (−0.582 − 1.00i)23-s + 0.464·27-s − 1.45·29-s + (−0.0743 + 0.128i)31-s + (0.0298 + 0.0517i)33-s + (0.464 + 0.805i)37-s + (−0.0663 + 0.114i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.0725 - 0.997i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.015780038\)
\(L(\frac12)\) \(\approx\) \(1.015780038\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 2.09i)T \)
good3 \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.414 - 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.82 - 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.79 + 4.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.82T + 29T^{2} \)
31 \( 1 + (0.414 - 0.717i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.82T + 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 + (-5.82 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.82 - 4.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.32 + 5.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.44 - 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-1.82 + 3.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 + (-2.67 - 4.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.893395574041003305068751563994, −9.018542534061322014711495645505, −7.84082619815839072373536256098, −7.58954143660168968696931818755, −6.51369718810926880530182727456, −5.79157157349672635529882034176, −4.47393608699894926136019936824, −3.93606414936728860722471152184, −2.57763522306340093920169065299, −1.49115930108142756793071327981, 0.39803953002408931483369339654, 2.27777779822076346478970234691, 3.13637838588076258290975611670, 4.17760411041282081733129406371, 5.22616166657849796455177735601, 5.98467260816422169940636112643, 7.06608519934883792704180485853, 7.49303348968151024107851414602, 9.053608552144786046255044478242, 9.205759219156756039035177310484

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