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Properties

Label 2-1400-280.187-c0-0-0
Degree $2$
Conductor $1400$
Sign $0.629 - 0.777i$
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 + (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 + (0.258 − 0.965i)18-s + (−0.866 + 1.5i)19-s + (−0.707 + 0.707i)22-s + (−0.965 + 0.258i)23-s + (1.49 + 0.866i)26-s + (0.258 − 0.965i)28-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 + (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 + (0.258 − 0.965i)18-s + (−0.866 + 1.5i)19-s + (−0.707 + 0.707i)22-s + (−0.965 + 0.258i)23-s + (1.49 + 0.866i)26-s + (0.258 − 0.965i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.629 - 0.777i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.629 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5620796839\)
\(L(\frac12)\) \(\approx\) \(0.5620796839\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.980945383276382793122784162452, −9.313790755728625588006542667923, −8.370347839669004128628897656994, −7.74767436457111810978445648998, −6.62353183629219428097570880985, −5.60842800597663977134185127713, −4.58446139587536299436622048801, −3.76622246855577697835317856075, −2.59671748629072887552206970704, −1.79225768456469176711182296686, 0.49481425394406403515591099010, 2.44921292152753361691213962454, 3.96842703862419395729650348543, 4.61585513625493121256379346295, 5.59955105890198956630343278661, 6.60505470979120142568139332634, 7.35504194044617833342464751957, 7.60080259761014752581051192604, 8.912902177744506787993302921633, 9.582906992639026686858382771164

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