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Properties

Label 2-1400-280.243-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.413 + 0.910i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 + 1.22i)13-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)18-s + (0.866 − 1.5i)19-s + (−0.707 − 0.707i)22-s + (0.258 + 0.965i)23-s + (1.49 + 0.866i)26-s + (−0.965 − 0.258i)28-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (1.22 + 1.22i)13-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)18-s + (0.866 − 1.5i)19-s + (−0.707 − 0.707i)22-s + (0.258 + 0.965i)23-s + (1.49 + 0.866i)26-s + (−0.965 − 0.258i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.413 + 0.910i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.413 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.785624655\)
\(L(\frac12)\) \(\approx\) \(1.785624655\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.575029665601527585430174953799, −9.028576639136068214474361478021, −7.85107638623922308995399758381, −6.80468829872123413329455006768, −6.30509094748128328385133804064, −5.46880447918343853166473304128, −4.42169588783792847640414017993, −3.38833353448210888562074158099, −2.94604232055429392529926994202, −1.19156858487768440120641321874, 2.07627177707208632520451503859, 3.06182138894486125860872480624, 3.77007004961947542021388278941, 5.23665541381007580774276546439, 5.58103900917705800769833585095, 6.37406913884182866299088483637, 7.42297228160502103659396119577, 8.200078503996262217395118893195, 8.841155584502329454763613787759, 10.24246957953630691676952823059

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