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Properties

Label 2-1400-280.27-c0-0-0
Degree $2$
Conductor $1400$
Sign $0.437 + 0.899i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s − i·9-s + 11-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.965 + 0.258i)18-s + (−0.258 − 0.965i)22-s + (0.707 + 0.707i)23-s + (0.258 − 0.965i)28-s + 1.73·29-s + (−0.965 − 0.258i)32-s + (0.499 + 0.866i)36-s + (0.707 − 0.707i)37-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s − i·9-s + 11-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.965 + 0.258i)18-s + (−0.258 − 0.965i)22-s + (0.707 + 0.707i)23-s + (0.258 − 0.965i)28-s + 1.73·29-s + (−0.965 − 0.258i)32-s + (0.499 + 0.866i)36-s + (0.707 − 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8717698757\)
\(L(\frac12)\) \(\approx\) \(0.8717698757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
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

−9.591017535837172252277672686647, −9.016981767247918872383334069585, −8.448800303771506850548183860152, −7.16012246128193704093702922799, −6.35568151144864724460359912368, −5.38857264802007331951353534908, −4.16808792919538325138385617295, −3.42009768972438794544522950462, −2.51535798108737488294081546008, −1.06478779223627187350958146069, 1.17540663682015430708949277025, 2.97566397800435230888191232088, 4.29920894577602660242770538656, 4.79597496847760369818627823695, 6.13056812671936432675225697800, 6.58860756124217453162568073014, 7.47081019771120729411830366085, 8.156794603937848531467685689899, 9.054355706832849009406590933507, 9.735764320973501097836460318163

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