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Properties

Label 2-1400-35.4-c1-0-7
Degree $2$
Conductor $1400$
Sign $0.732 - 0.680i$
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  + (−0.403 − 0.232i)3-s + (−2.02 − 1.70i)7-s + (−1.39 − 2.41i)9-s + (−0.688 + 1.19i)11-s + 5.12i·13-s + (0.292 + 0.168i)17-s + (2.99 + 5.19i)19-s + (0.420 + 1.15i)21-s + (−5.67 + 3.27i)23-s + 2.69i·27-s + 7.99·29-s + (3.62 − 6.27i)31-s + (0.555 − 0.320i)33-s + (5.24 − 3.02i)37-s + (1.19 − 2.06i)39-s + ⋯
L(s)  = 1  + (−0.232 − 0.134i)3-s + (−0.765 − 0.643i)7-s + (−0.463 − 0.803i)9-s + (−0.207 + 0.359i)11-s + 1.42i·13-s + (0.0709 + 0.0409i)17-s + (0.687 + 1.19i)19-s + (0.0916 + 0.252i)21-s + (−1.18 + 0.682i)23-s + 0.518i·27-s + 1.48·29-s + (0.650 − 1.12i)31-s + (0.0966 − 0.0558i)33-s + (0.861 − 0.497i)37-s + (0.190 − 0.330i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.732 - 0.680i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.110665108\)
\(L(\frac12)\) \(\approx\) \(1.110665108\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.02 + 1.70i)T \)
good3 \( 1 + (0.403 + 0.232i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.688 - 1.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.12iT - 13T^{2} \)
17 \( 1 + (-0.292 - 0.168i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.99 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.67 - 3.27i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.99T + 29T^{2} \)
31 \( 1 + (-3.62 + 6.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.24 + 3.02i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.68T + 41T^{2} \)
43 \( 1 - 5.02iT - 43T^{2} \)
47 \( 1 + (-5.75 + 3.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.21 - 2.43i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.301 + 0.521i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.54 - 11.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.4 + 6.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.39T + 71T^{2} \)
73 \( 1 + (1.45 + 0.842i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.80 - 6.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.49iT - 83T^{2} \)
89 \( 1 + (6.19 + 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.691iT - 97T^{2} \)
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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

−9.724787134624988952228137658116, −9.030808441311660634152392736215, −7.919019937785918891361713089155, −7.18688454173246911109118961239, −6.27714590357141917476476027581, −5.83050730296570175607511456920, −4.34272001800731827096300640291, −3.76996650490672673939816775749, −2.54127897470073506887502577919, −1.06033016127195398514656537600, 0.55828857053670581395585736037, 2.59948409497420642366653315580, 3.02802207477264462211094941558, 4.52563049263927381950269625611, 5.42381968204662551306315374260, 5.99162837959487979926675695962, 6.96026212918183400616749455772, 8.087113572262290240687549721854, 8.502864646686909570737118760481, 9.552380195785829657335277863322

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