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Properties

Label 2-1400-35.9-c1-0-19
Degree $2$
Conductor $1400$
Sign $0.981 - 0.192i$
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  + (−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: $0.981 - 0.192i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (849, \cdot )$
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

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

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