Properties

Label 2-1400-35.4-c1-0-19
Degree $2$
Conductor $1400$
Sign $0.330 - 0.943i$
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  + (2.59 + 1.5i)3-s + (1.73 + 2i)7-s + (3 + 5.19i)9-s + (0.5 − 0.866i)11-s − 2i·13-s + (−2.59 − 1.5i)17-s + (2.5 + 4.33i)19-s + (1.5 + 7.79i)21-s + (2.59 − 1.5i)23-s + 9i·27-s + 6·29-s + (0.5 − 0.866i)31-s + (2.59 − 1.5i)33-s + (−4.33 + 2.5i)37-s + (3 − 5.19i)39-s + ⋯
L(s)  = 1  + (1.49 + 0.866i)3-s + (0.654 + 0.755i)7-s + (1 + 1.73i)9-s + (0.150 − 0.261i)11-s − 0.554i·13-s + (−0.630 − 0.363i)17-s + (0.573 + 0.993i)19-s + (0.327 + 1.70i)21-s + (0.541 − 0.312i)23-s + 1.73i·27-s + 1.11·29-s + (0.0898 − 0.155i)31-s + (0.452 − 0.261i)33-s + (−0.711 + 0.410i)37-s + (0.480 − 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.077758044\)
\(L(\frac12)\) \(\approx\) \(3.077758044\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.52 + 5.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (-6.06 - 3.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−9.621338096074645903857884652626, −8.794584773090210978916035046766, −8.331350464505880152302142327411, −7.72095486495764131996988297336, −6.49682826297776569943999222570, −5.22996115172301160884515969663, −4.62189654409106141935307866709, −3.47061251594437946512355006275, −2.81698045225542291486480440355, −1.73749201477878856878589822295, 1.17823540563910402632304150023, 2.09541406879392420299476405459, 3.13553626190755665717283993062, 4.09117886390434396199635671697, 5.01143669925527556259083708861, 6.66382406787934641398845607328, 7.03810457015652880763639097212, 7.81207659145143750155855698150, 8.600194310859169897894001267187, 9.094044549603208368503540009519

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