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Properties

Label 2-1400-35.9-c1-0-16
Degree $2$
Conductor $1400$
Sign $0.946 + 0.324i$
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  + (−1.73 + i)3-s + (−2.59 + 0.5i)7-s + (0.499 − 0.866i)9-s + (−2 − 3.46i)11-s + 2i·13-s + (−2.59 + 1.5i)17-s + (4 − 3.46i)21-s + (2.59 + 1.5i)23-s − 4.00i·27-s + 6·29-s + (−4.5 − 7.79i)31-s + (6.92 + 3.99i)33-s + (−2 − 3.46i)39-s + 5·41-s + 6i·43-s + ⋯
L(s)  = 1  + (−0.999 + 0.577i)3-s + (−0.981 + 0.188i)7-s + (0.166 − 0.288i)9-s + (−0.603 − 1.04i)11-s + 0.554i·13-s + (−0.630 + 0.363i)17-s + (0.872 − 0.755i)21-s + (0.541 + 0.312i)23-s − 0.769i·27-s + 1.11·29-s + (−0.808 − 1.39i)31-s + (1.20 + 0.696i)33-s + (−0.320 − 0.554i)39-s + 0.780·41-s + 0.914i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.946 + 0.324i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.946 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6673901678\)
\(L(\frac12)\) \(\approx\) \(0.6673901678\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 0.5i)T \)
good3 \( 1 + (1.73 - i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (2.59 - 1.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (7.79 + 4.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.1 + 7i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11T + 71T^{2} \)
73 \( 1 + (-1.73 + i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-5.5 + 9.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11iT - 97T^{2} \)
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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

−9.654028258557820495140992808530, −8.856833062513436797012709436849, −7.975737084382127609758235409530, −6.78475213600872791645550723338, −6.10049209902807316810054933129, −5.48015728399402665245914785877, −4.52460607481560514787712500673, −3.54469443243529773928216285594, −2.43661604650621325620567295816, −0.44812490500234005034926079352, 0.798065529506043271744585590068, 2.37382794954065898906508732672, 3.48211116493203785859396962733, 4.81601773606973277345984539694, 5.45185554225357682759878629799, 6.61206436798506560101778272119, 6.82235096945781159743068276381, 7.76887977261897948621842478368, 8.882325115164058840942700521335, 9.715516755134187242146608923434

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