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Properties

Label 2-1400-35.13-c1-0-16
Degree $2$
Conductor $1400$
Sign $0.977 + 0.211i$
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  + (−1.41 + 1.41i)3-s + (−2.64 + 0.0496i)7-s − 0.987i·9-s − 5.75·11-s + (2.89 − 2.89i)13-s + (−2.13 − 2.13i)17-s + 2.18·19-s + (3.66 − 3.80i)21-s + (4.79 + 4.79i)23-s + (−2.84 − 2.84i)27-s + 5.19i·29-s − 6.68i·31-s + (8.12 − 8.12i)33-s + (6.50 − 6.50i)37-s + 8.17i·39-s + ⋯
L(s)  = 1  + (−0.815 + 0.815i)3-s + (−0.999 + 0.0187i)7-s − 0.329i·9-s − 1.73·11-s + (0.802 − 0.802i)13-s + (−0.517 − 0.517i)17-s + 0.502·19-s + (0.799 − 0.830i)21-s + (1.00 + 1.00i)23-s + (−0.546 − 0.546i)27-s + 0.964i·29-s − 1.20i·31-s + (1.41 − 1.41i)33-s + (1.06 − 1.06i)37-s + 1.30i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7356892520\)
\(L(\frac12)\) \(\approx\) \(0.7356892520\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0496i)T \)
good3 \( 1 + (1.41 - 1.41i)T - 3iT^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 + (-2.89 + 2.89i)T - 13iT^{2} \)
17 \( 1 + (2.13 + 2.13i)T + 17iT^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + (-4.79 - 4.79i)T + 23iT^{2} \)
29 \( 1 - 5.19iT - 29T^{2} \)
31 \( 1 + 6.68iT - 31T^{2} \)
37 \( 1 + (-6.50 + 6.50i)T - 37iT^{2} \)
41 \( 1 + 0.846iT - 41T^{2} \)
43 \( 1 + (2.68 + 2.68i)T + 43iT^{2} \)
47 \( 1 + (-4.55 - 4.55i)T + 47iT^{2} \)
53 \( 1 + (-0.750 - 0.750i)T + 53iT^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 - 7.62iT - 61T^{2} \)
67 \( 1 + (-2.14 + 2.14i)T - 67iT^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 + (3.51 - 3.51i)T - 73iT^{2} \)
79 \( 1 - 1.15iT - 79T^{2} \)
83 \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \)
89 \( 1 - 5.57T + 89T^{2} \)
97 \( 1 + (5.66 + 5.66i)T + 97iT^{2} \)
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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

−9.703650193686295456316465100704, −8.955214644822736776272764296931, −7.81694189002569387917182344409, −7.13818519137769342229857768723, −5.77880566041903120004170942237, −5.57687077480986897501164360184, −4.59628999705929257897261690173, −3.45938817127399912383880803289, −2.60276225007362606937431811103, −0.46366286570575012916240960146, 0.840867463867381853565716289816, 2.34582207801645128726998715158, 3.39771459270811180207532369979, 4.69431441768041620122895102782, 5.63948920494785124738262459442, 6.45220993789432124145297439417, 6.86378072894354745365907320630, 7.88098909594958343757151579185, 8.717673718568431495718228603468, 9.657897622957911835082925073138

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