Properties

Degree $2$
Conductor $1400$
Sign $0.981 + 0.192i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (1.73 − 2i)7-s + (−1 − 1.73i)9-s + (−1.5 + 2.59i)11-s + 6i·13-s + (4.33 + 2.5i)17-s + (0.5 + 0.866i)19-s + (−2.5 + 0.866i)21-s + (6.06 − 3.5i)23-s + 5i·27-s − 2·29-s + (2.5 − 4.33i)31-s + (2.59 − 1.5i)33-s + (2.59 − 1.5i)37-s + (3 − 5.19i)39-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.654 − 0.755i)7-s + (−0.333 − 0.577i)9-s + (−0.452 + 0.783i)11-s + 1.66i·13-s + (1.05 + 0.606i)17-s + (0.114 + 0.198i)19-s + (−0.545 + 0.188i)21-s + (1.26 − 0.729i)23-s + 0.962i·27-s − 0.371·29-s + (0.449 − 0.777i)31-s + (0.452 − 0.261i)33-s + (0.427 − 0.246i)37-s + (0.480 − 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.981 + 0.192i$
Motivic weight: \(1\)
Character: $\chi_{1400} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.981 + 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.458127481\)
\(L(\frac12)\) \(\approx\) \(1.458127481\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 2i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + (-4.33 - 2.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.06 + 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-4.33 + 2.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 - 4.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-6.06 - 3.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.605050039842320847910643789940, −8.730954247825985294285021742219, −7.79234262467323348165272709373, −7.02971051891979341341918578724, −6.40972783740450337227118053151, −5.33276902726813355436507377495, −4.49627071815451384743796376604, −3.61853060528631916355184799554, −2.11373911164623239737350057742, −0.962581114796030014492131240450, 0.883185381773031845670721990047, 2.60157821688622295031232664876, 3.31526122371609891395444245742, 4.96784455283808950449597526500, 5.35604468720461207669288080531, 5.89742857156455839942674100486, 7.33980197886924852336474234111, 8.054471182996292269382879158410, 8.634834977772532188926940137428, 9.674245725966868755328527607372

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