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Properties

Label 2-1400-280.69-c0-0-3
Degree $2$
Conductor $1400$
Sign $0.894 + 0.447i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·7-s − 0.999i·8-s + 9-s + 1.73i·11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)18-s + (0.866 + 1.49i)22-s − i·23-s + (0.866 + 0.499i)28-s − 1.73i·29-s + (−0.866 − 0.499i)32-s + (0.499 − 0.866i)36-s − 1.73·37-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·7-s − 0.999i·8-s + 9-s + 1.73i·11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)18-s + (0.866 + 1.49i)22-s − i·23-s + (0.866 + 0.499i)28-s − 1.73i·29-s + (−0.866 − 0.499i)32-s + (0.499 − 0.866i)36-s − 1.73·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.867192041\)
\(L(\frac12)\) \(\approx\) \(1.867192041\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.969759696331736670622159253965, −9.184758723122127948815400066823, −8.006380723640101682683280881690, −6.95326472785480355241422688688, −6.44221876745650083029023119916, −5.23719777479881426953485330547, −4.65668643985632725499620215615, −3.78360694597269156595348377952, −2.42266313709867637572671166885, −1.77614059708378109395401590793, 1.52034457622724447022130108464, 3.33390983473758282201081445631, 3.67399747289231948260723593332, 4.85128989002180292285306507561, 5.57637640297452060270368011031, 6.70527757626231824758033501349, 7.09820131387854808398554814410, 8.070352244532070642464511138615, 8.740036030852864173957089676474, 9.924714478927758174527957209725

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