Properties

Label 2-1400-5.4-c1-0-21
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  + 2.56i·3-s i·7-s − 3.56·9-s − 2.56·11-s − 5.68i·13-s − 3.43i·17-s − 1.12·19-s + 2.56·21-s − 5.12i·23-s − 1.43i·27-s − 4.56·29-s − 10.2·31-s − 6.56i·33-s − 8.24i·37-s + 14.5·39-s + ⋯
L(s)  = 1  + 1.47i·3-s − 0.377i·7-s − 1.18·9-s − 0.772·11-s − 1.57i·13-s − 0.833i·17-s − 0.257·19-s + 0.558·21-s − 1.06i·23-s − 0.276i·27-s − 0.847·29-s − 1.84·31-s − 1.14i·33-s − 1.35i·37-s + 2.33·39-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

\(L(1)\) \(\approx\) \(0.8001876182\)
\(L(\frac12)\) \(\approx\) \(0.8001876182\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56iT - 3T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
13 \( 1 + 5.68iT - 13T^{2} \)
17 \( 1 + 3.43iT - 17T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 8.24iT - 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 - 1.12iT - 43T^{2} \)
47 \( 1 + 6.56iT - 47T^{2} \)
53 \( 1 + 4.87iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 15.1T + 61T^{2} \)
67 \( 1 - 14.2iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 3.12T + 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−9.531572538859434418605456116859, −8.821262770558197284132132985258, −7.88080327395028406212267543543, −7.13355210604582170156060712752, −5.64975068822626243874603517801, −5.27824651757590745895446837867, −4.26622302712820524244539116873, −3.46915940855910231562766083746, −2.51305106345929472698487532338, −0.30958258985644181039084765392, 1.57004024807407151026381495170, 2.16665484042412804600039542478, 3.49295853126026088006959778651, 4.75128985922686082330200039107, 5.91100423073670789324109130592, 6.40493314811282025128482289080, 7.47822236681689997123618363718, 7.74630746231366395915718045742, 8.886640623389109044579893763625, 9.412655408889341252374735187996

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