Properties

Label 2-1400-280.227-c0-0-0
Degree $2$
Conductor $1400$
Sign $0.413 - 0.910i$
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  + (−0.965 − 0.258i)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 + (0.965 − 0.258i)18-s + (0.866 + 1.5i)19-s + (0.707 − 0.707i)22-s + (−0.258 + 0.965i)23-s + (1.49 − 0.866i)26-s + (0.965 − 0.258i)28-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)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 + (0.965 − 0.258i)18-s + (0.866 + 1.5i)19-s + (0.707 − 0.707i)22-s + (−0.258 + 0.965i)23-s + (1.49 − 0.866i)26-s + (0.965 − 0.258i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.413 - 0.910i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.413 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5719567295\)
\(L(\frac12)\) \(\approx\) \(0.5719567295\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
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

−9.880390038394938079558133895801, −9.258955773612922295256794386199, −8.169816396613874052430248934658, −7.58508719647693763513763242294, −7.13302604869774877239464138853, −5.84396185942796397923837095049, −4.85387813203063674462403538027, −3.79061083581996722743594145270, −2.45745800860247170163570861473, −1.63034855605937855165070787651, 0.63045122520183065317989600293, 2.53544836201763875529287668594, 2.94433072391835543899915454449, 4.98980739210842438544548205804, 5.51856783783194984091914286084, 6.37285307866907345167183001636, 7.44446388244459279016202196392, 8.162343436641977506667578916062, 8.690168023703550907243038944130, 9.479198129631908431429591203627

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