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Properties

Label 2-2800-140.87-c0-0-1
Degree $2$
Conductor $2800$
Sign $-0.635 + 0.772i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
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)3-s + (−0.965 + 0.258i)7-s + (0.499 + 0.866i)21-s + (0.448 − 1.67i)23-s + (−0.707 − 0.707i)27-s − i·29-s − 1.73i·41-s + (−1.22 − 1.22i)43-s + (−0.517 + 1.93i)47-s + (0.866 − 0.499i)49-s + (−1.5 + 0.866i)61-s + (−0.448 − 1.67i)67-s − 1.73·69-s + (−0.5 + 0.866i)81-s + (0.707 − 0.707i)83-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (−0.965 + 0.258i)7-s + (0.499 + 0.866i)21-s + (0.448 − 1.67i)23-s + (−0.707 − 0.707i)27-s − i·29-s − 1.73i·41-s + (−1.22 − 1.22i)43-s + (−0.517 + 1.93i)47-s + (0.866 − 0.499i)49-s + (−1.5 + 0.866i)61-s + (−0.448 − 1.67i)67-s − 1.73·69-s + (−0.5 + 0.866i)81-s + (0.707 − 0.707i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.635 + 0.772i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (2607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ -0.635 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8019464775\)
\(L(\frac12)\) \(\approx\) \(0.8019464775\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.965 - 0.258i)T \)
good3 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
47 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.724971879663253451510684535255, −7.80394724396226442760709316706, −7.10956972639880081438356928718, −6.39238600271335756823349618837, −6.01048229321641547187634849533, −4.86496016502335614178151530062, −3.87721878567414390170113052048, −2.84024431915636895926348958971, −1.93129060782538029649449096648, −0.53009888802168640512955650552, 1.53486688040036659608923374664, 3.12660354436648745382619911092, 3.59325593483899162959869441535, 4.62048914932903304726647929415, 5.25735564197068054383212776106, 6.15184911123688167722034587325, 6.96471018346374353185195329139, 7.66741989797693394216778283507, 8.705821493243430401108013686254, 9.504383887283012290684849057175

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