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Properties

Label 2-1400-1400.1021-c0-0-2
Degree $2$
Conductor $1400$
Sign $-0.535 - 0.844i$
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  + (−0.309 + 0.951i)2-s + (0.951 + 0.690i)3-s + (−0.809 − 0.587i)4-s + (0.951 + 0.309i)5-s + (−0.951 + 0.690i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.118 + 0.363i)9-s + (−0.587 + 0.809i)10-s + (−0.363 − 1.11i)12-s + (0.363 + 1.11i)13-s + (0.309 − 0.951i)14-s + (0.690 + 0.951i)15-s + (0.309 + 0.951i)16-s − 0.381·18-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (0.951 + 0.690i)3-s + (−0.809 − 0.587i)4-s + (0.951 + 0.309i)5-s + (−0.951 + 0.690i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.118 + 0.363i)9-s + (−0.587 + 0.809i)10-s + (−0.363 − 1.11i)12-s + (0.363 + 1.11i)13-s + (0.309 − 0.951i)14-s + (0.690 + 0.951i)15-s + (0.309 + 0.951i)16-s − 0.381·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237985582\)
\(L(\frac12)\) \(\approx\) \(1.237985582\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + T \)
good3 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.604422884980648343959346031350, −9.358911774202292239621905432344, −8.652279338514046644531358287760, −7.64129654338164140827816563586, −6.64735825733208648623634716660, −6.17230660245597089431921001629, −5.16858061631345423402404010487, −4.00035878326675919081495598049, −3.25838588822926833516245191684, −1.86298532243176376511015882646, 1.12233116392352343792814609319, 2.43838579966832259913146502127, 2.84926056531156137955195571858, 3.99046165098812142903869231700, 5.27702932287009874704040775481, 6.24134601791209941287710738719, 7.27179073309420620301147341253, 8.238059161897274028430707271024, 8.740526051401407335635526577438, 9.445312590662480514219143895337

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