Properties

Label 2-1400-35.27-c1-0-27
Degree $2$
Conductor $1400$
Sign $0.884 + 0.465i$
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  + (0.923 + 0.923i)3-s + (0.915 − 2.48i)7-s − 1.29i·9-s + 1.21·11-s + (0.996 + 0.996i)13-s + (0.567 − 0.567i)17-s + 0.103·19-s + (3.13 − 1.44i)21-s + (−1.43 + 1.43i)23-s + (3.96 − 3.96i)27-s − 4.97i·29-s − 4.35i·31-s + (1.12 + 1.12i)33-s + (2.54 + 2.54i)37-s + 1.83i·39-s + ⋯
L(s)  = 1  + (0.532 + 0.532i)3-s + (0.345 − 0.938i)7-s − 0.431i·9-s + 0.366·11-s + (0.276 + 0.276i)13-s + (0.137 − 0.137i)17-s + 0.0237·19-s + (0.684 − 0.315i)21-s + (−0.298 + 0.298i)23-s + (0.763 − 0.763i)27-s − 0.924i·29-s − 0.781i·31-s + (0.195 + 0.195i)33-s + (0.417 + 0.417i)37-s + 0.294i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.884 + 0.465i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.884 + 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.122570355\)
\(L(\frac12)\) \(\approx\) \(2.122570355\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.915 + 2.48i)T \)
good3 \( 1 + (-0.923 - 0.923i)T + 3iT^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 + (-0.996 - 0.996i)T + 13iT^{2} \)
17 \( 1 + (-0.567 + 0.567i)T - 17iT^{2} \)
19 \( 1 - 0.103T + 19T^{2} \)
23 \( 1 + (1.43 - 1.43i)T - 23iT^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 + 4.35iT - 31T^{2} \)
37 \( 1 + (-2.54 - 2.54i)T + 37iT^{2} \)
41 \( 1 + 7.06iT - 41T^{2} \)
43 \( 1 + (3.11 - 3.11i)T - 43iT^{2} \)
47 \( 1 + (-7.23 + 7.23i)T - 47iT^{2} \)
53 \( 1 + (0.882 - 0.882i)T - 53iT^{2} \)
59 \( 1 - 8.28T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \)
71 \( 1 - 0.329T + 71T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 - 13.6iT - 79T^{2} \)
83 \( 1 + (-4.61 - 4.61i)T + 83iT^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (2.40 - 2.40i)T - 97iT^{2} \)
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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

−9.699559852258327439529137020515, −8.689180554531742569249947560531, −8.029594922439226675404286764043, −7.08632793210726222542264384405, −6.32458832568693589615332130058, −5.19347556820502998202985147780, −4.00274905916682925273207175587, −3.78893221725632158354809792327, −2.37131987550211202828234901402, −0.897586441396017728825779646072, 1.42048913915827499452254530056, 2.39803532243645805832253958542, 3.32396661360207236250371889129, 4.63853118528633590184488755515, 5.49123123048758643075087310942, 6.38303649473501196600324545382, 7.32800181321314910295798552675, 8.131683229277186096891160789003, 8.675835660988076945556957620517, 9.397374112851073309938776637309

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