Properties

Label 2-1400-35.13-c1-0-27
Degree $2$
Conductor $1400$
Sign $-0.220 + 0.975i$
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 + (−2.48 − 0.915i)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 + (−0.938 − 0.345i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4168272456\)
\(L(\frac12)\) \(\approx\) \(0.4168272456\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.48 + 0.915i)T \)
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

−9.527867041609732749456303237377, −8.674295610067667499148271409855, −7.60668807970463525083783753590, −6.80851495798383290195538466077, −6.01950249781205990733121868313, −5.10455722885609861412326199590, −4.24563097163799463555979708091, −3.36069156608663259702263297291, −2.04831897769660018929444416825, −0.18993483627494238534008572391, 1.26831193487252739378758187129, 2.73965641562164256098725989506, 3.67559329968801363044146545419, 4.84384784069474724469315357763, 6.03767544012496378393115817091, 6.31892955576346338708799773038, 7.21241554590611583123012227648, 8.114925567792465624278288151247, 9.147974125168837077346968727281, 9.700004206065555043673158124143

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