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Properties

Label 2-1400-35.27-c1-0-34
Degree $2$
Conductor $1400$
Sign $-0.998 - 0.0497i$
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  + (−0.409 − 0.409i)3-s + (2.54 − 0.738i)7-s − 2.66i·9-s − 3.54·11-s + (−2.95 − 2.95i)13-s + (−5.59 + 5.59i)17-s − 3.59·19-s + (−1.34 − 0.737i)21-s + (0.0472 − 0.0472i)23-s + (−2.31 + 2.31i)27-s + 5.34i·29-s − 10.3i·31-s + (1.45 + 1.45i)33-s + (7.80 + 7.80i)37-s + 2.41i·39-s + ⋯
L(s)  = 1  + (−0.236 − 0.236i)3-s + (0.960 − 0.278i)7-s − 0.888i·9-s − 1.07·11-s + (−0.818 − 0.818i)13-s + (−1.35 + 1.35i)17-s − 0.824·19-s + (−0.292 − 0.160i)21-s + (0.00986 − 0.00986i)23-s + (−0.446 + 0.446i)27-s + 0.993i·29-s − 1.86i·31-s + (0.252 + 0.252i)33-s + (1.28 + 1.28i)37-s + 0.386i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.998 - 0.0497i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.998 - 0.0497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3352517557\)
\(L(\frac12)\) \(\approx\) \(0.3352517557\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 + 0.738i)T \)
good3 \( 1 + (0.409 + 0.409i)T + 3iT^{2} \)
11 \( 1 + 3.54T + 11T^{2} \)
13 \( 1 + (2.95 + 2.95i)T + 13iT^{2} \)
17 \( 1 + (5.59 - 5.59i)T - 17iT^{2} \)
19 \( 1 + 3.59T + 19T^{2} \)
23 \( 1 + (-0.0472 + 0.0472i)T - 23iT^{2} \)
29 \( 1 - 5.34iT - 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + (-7.80 - 7.80i)T + 37iT^{2} \)
41 \( 1 + 5.63iT - 41T^{2} \)
43 \( 1 + (6.93 - 6.93i)T - 43iT^{2} \)
47 \( 1 + (3.44 - 3.44i)T - 47iT^{2} \)
53 \( 1 + (-0.646 + 0.646i)T - 53iT^{2} \)
59 \( 1 + 9.22T + 59T^{2} \)
61 \( 1 + 3.51iT - 61T^{2} \)
67 \( 1 + (1.70 + 1.70i)T + 67iT^{2} \)
71 \( 1 + 8.54T + 71T^{2} \)
73 \( 1 + (2.43 + 2.43i)T + 73iT^{2} \)
79 \( 1 + 5.72iT - 79T^{2} \)
83 \( 1 + (2.04 + 2.04i)T + 83iT^{2} \)
89 \( 1 - 8.18T + 89T^{2} \)
97 \( 1 + (6.23 - 6.23i)T - 97iT^{2} \)
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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

−9.089147253888561166895176665031, −8.151868456921288424295754761626, −7.72867922716256434678667720853, −6.62521753601709050847143894490, −5.91498035409328392722994427708, −4.87919208035860505786453473664, −4.15948756008448552665345938417, −2.82738591720294852370819838123, −1.70646866022045643206245659284, −0.12830617763390862700091743132, 2.02287985380198908771155862564, 2.62728026584577022442830145705, 4.50302591672537156608230146364, 4.75953625033441796008761151920, 5.61551077563664771693571013832, 6.85453360914377599269995216443, 7.56343887856968364537744639415, 8.371362377110089119352871340603, 9.105316759411328639112331649794, 10.07254017252179908094877198777

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