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Properties

Label 2-1400-5.4-c1-0-12
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  + 1.56i·3-s + i·7-s + 0.561·9-s + 1.56·11-s − 6.68i·13-s + 7.56i·17-s + 7.12·19-s − 1.56·21-s − 3.12i·23-s + 5.56i·27-s − 0.438·29-s + 6.24·31-s + 2.43i·33-s − 8.24i·37-s + 10.4·39-s + ⋯
L(s)  = 1  + 0.901i·3-s + 0.377i·7-s + 0.187·9-s + 0.470·11-s − 1.85i·13-s + 1.83i·17-s + 1.63·19-s − 0.340·21-s − 0.651i·23-s + 1.07i·27-s − 0.0814·29-s + 1.12·31-s + 0.424i·33-s − 1.35i·37-s + 1.67·39-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

\(L(1)\) \(\approx\) \(1.866959930\)
\(L(\frac12)\) \(\approx\) \(1.866959930\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56iT - 3T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
13 \( 1 + 6.68iT - 13T^{2} \)
17 \( 1 - 7.56iT - 17T^{2} \)
19 \( 1 - 7.12T + 19T^{2} \)
23 \( 1 + 3.12iT - 23T^{2} \)
29 \( 1 + 0.438T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + 8.24iT - 37T^{2} \)
41 \( 1 + 1.12T + 41T^{2} \)
43 \( 1 - 7.12iT - 43T^{2} \)
47 \( 1 - 2.43iT - 47T^{2} \)
53 \( 1 - 13.1iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 6.87T + 61T^{2} \)
67 \( 1 - 2.24iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 4.24iT - 73T^{2} \)
79 \( 1 + 0.684T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 - 1.31iT - 97T^{2} \)
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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

−9.810271795354044881577996470407, −9.008627381858704793270909788688, −8.151056235889374734970644333664, −7.45512440668321528604379298256, −6.16134918238568241350375035505, −5.55387426926461983209516831355, −4.59427708723495744443373766246, −3.68352557202805543211790362009, −2.85968825103036031012714762032, −1.21085809393547864215875895238, 0.951819091804065375769124304012, 1.93643804931873142741763856806, 3.21512375716778850391782945953, 4.36550296875690537420777411144, 5.18986133889842048722558690371, 6.51148482547309782437921226605, 7.01129556534427408274517385793, 7.50999027318510673404282903223, 8.586363988842735925539568463532, 9.605410669864057035833094951702

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