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Properties

Label 2-1400-35.27-c1-0-32
Degree $2$
Conductor $1400$
Sign $-0.996 + 0.0776i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.848 − 0.848i)3-s + (0.406 − 2.61i)7-s − 1.56i·9-s − 2.56·11-s + (2.17 + 2.17i)13-s + (2.17 − 2.17i)17-s − 8.54·19-s + (−2.56 + 1.87i)21-s + (5.03 − 5.03i)23-s + (−3.86 + 3.86i)27-s − 5.68i·29-s + 4.79i·31-s + (2.17 + 2.17i)33-s + (−2.82 − 2.82i)37-s − 3.68i·39-s + ⋯
L(s)  = 1  + (−0.489 − 0.489i)3-s + (0.153 − 0.988i)7-s − 0.520i·9-s − 0.772·11-s + (0.602 + 0.602i)13-s + (0.526 − 0.526i)17-s − 1.95·19-s + (−0.558 + 0.408i)21-s + (1.05 − 1.05i)23-s + (−0.744 + 0.744i)27-s − 1.05i·29-s + 0.861i·31-s + (0.378 + 0.378i)33-s + (−0.464 − 0.464i)37-s − 0.590i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.996 + 0.0776i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.996 + 0.0776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6786439148\)
\(L(\frac12)\) \(\approx\) \(0.6786439148\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.406 + 2.61i)T \)
good3 \( 1 + (0.848 + 0.848i)T + 3iT^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
13 \( 1 + (-2.17 - 2.17i)T + 13iT^{2} \)
17 \( 1 + (-2.17 + 2.17i)T - 17iT^{2} \)
19 \( 1 + 8.54T + 19T^{2} \)
23 \( 1 + (-5.03 + 5.03i)T - 23iT^{2} \)
29 \( 1 + 5.68iT - 29T^{2} \)
31 \( 1 - 4.79iT - 31T^{2} \)
37 \( 1 + (2.82 + 2.82i)T + 37iT^{2} \)
41 \( 1 - 8.54iT - 41T^{2} \)
43 \( 1 + (0.620 - 0.620i)T - 43iT^{2} \)
47 \( 1 + (0.848 - 0.848i)T - 47iT^{2} \)
53 \( 1 + (4.41 - 4.41i)T - 53iT^{2} \)
59 \( 1 + 3.74T + 59T^{2} \)
61 \( 1 + 4.79iT - 61T^{2} \)
67 \( 1 + (2.20 + 2.20i)T + 67iT^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + (-3.39 - 3.39i)T + 73iT^{2} \)
79 \( 1 + 8.80iT - 79T^{2} \)
83 \( 1 + (5.66 + 5.66i)T + 83iT^{2} \)
89 \( 1 + 4.79T + 89T^{2} \)
97 \( 1 + (-13.3 + 13.3i)T - 97iT^{2} \)
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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

−9.108517264585304664223830257534, −8.308659855669929746226864961441, −7.43174835283649076286446094361, −6.62941489276831138354159552670, −6.14422191444863580643489602166, −4.87068737892295931758812129301, −4.13493216535744774352965396604, −2.95629192466490712256500606265, −1.52525088158962568921434034412, −0.29011712315015713448232922683, 1.81009038058949849473405599635, 2.91554603890090169787187871076, 4.08101449070885471865034167902, 5.22189964567545470224578798391, 5.54326013592427638523337948398, 6.50039353392159684133015956180, 7.71340115216059431111815789311, 8.395958077248843028899038812655, 9.070262892399304376105423642541, 10.17266805740193494643331832468

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