Properties

Label 2-1400-280.27-c0-0-5
Degree $2$
Conductor $1400$
Sign $-0.525 + 0.850i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s i·9-s − 2·11-s − 1.00i·14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (−1.41 + 1.41i)22-s + (1.41 + 1.41i)23-s + (−0.707 − 0.707i)28-s + (−0.707 + 0.707i)32-s − 1.00·36-s + (1.41 − 1.41i)37-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s i·9-s − 2·11-s − 1.00i·14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (−1.41 + 1.41i)22-s + (1.41 + 1.41i)23-s + (−0.707 − 0.707i)28-s + (−0.707 + 0.707i)32-s − 1.00·36-s + (1.41 − 1.41i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.525 + 0.850i$
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: \((2,\ 1400,\ (\ :0),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454543453\)
\(L(\frac12)\) \(\approx\) \(1.454543453\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.691748151295090939711516287963, −8.888717788292081697757836396209, −7.70120317507425053881318247578, −7.10921833043693498197934647900, −5.83622774236759075878055463790, −5.22282530444739120148876087949, −4.32983848716210785356525031360, −3.36824504198896216897302967310, −2.43216942156901137481886955249, −0.990213845625907065929441093178, 2.36079290218657648514300410497, 2.86991368829625827495916447409, 4.55386295984599860582520535068, 5.05297713927023676042057602166, 5.63856108595301854328853032195, 6.76176677786536511397659401339, 7.73798201724309262489502264937, 8.171039605770033485512851782016, 8.812836785592723978767507124967, 10.15086681854757701546933941973

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