Properties

Label 2-1400-5.4-c1-0-2
Degree $2$
Conductor $1400$
Sign $-0.447 + 0.894i$
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  + 3i·3-s + i·7-s − 6·9-s − 5·11-s + 5i·13-s − 7i·17-s + 2·19-s − 3·21-s + 2i·23-s − 9i·27-s − 7·29-s + 4·31-s − 15i·33-s − 6i·37-s − 15·39-s + ⋯
L(s)  = 1  + 1.73i·3-s + 0.377i·7-s − 2·9-s − 1.50·11-s + 1.38i·13-s − 1.69i·17-s + 0.458·19-s − 0.654·21-s + 0.417i·23-s − 1.73i·27-s − 1.29·29-s + 0.718·31-s − 2.61i·33-s − 0.986i·37-s − 2.40·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4729179979\)
\(L(\frac12)\) \(\approx\) \(0.4729179979\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - 3iT - 3T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 7iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 13iT - 97T^{2} \)
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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

−9.912945338922367994741340918009, −9.454077248532162809256220197409, −8.835040812046141637034691858472, −7.83595882214809795956506854805, −6.85775665142860311628507074144, −5.46012397543347808758332835704, −5.15342213409149017338338020428, −4.26048787228535012777427976783, −3.25862285975149404729416265583, −2.36356503599207999051583964960, 0.18358137541973819908023094126, 1.47810162955999494268069435192, 2.54634699089576332556645417472, 3.48323871218730750841209915753, 5.14392213520221523780011852313, 5.81848835328526736895234409555, 6.65987578942819927692899991328, 7.52092873245584717882288237667, 8.116839195717068430291532311998, 8.458500745950601494457365975183

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