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Properties

Label 2-2800-140.27-c0-0-1
Degree $2$
Conductor $2800$
Sign $-0.229 - 0.973i$
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  + (−1.41 + 1.41i)3-s + (−0.707 − 0.707i)7-s − 3.00i·9-s + 2.00·21-s + (2.82 + 2.82i)27-s + 2i·29-s + (1.41 + 1.41i)47-s + 1.00i·49-s + (−2.12 + 2.12i)63-s − 5.00·81-s + (1.41 − 1.41i)83-s + (−2.82 − 2.82i)87-s + (−1.41 + 1.41i)103-s + 2i·109-s + ⋯
L(s)  = 1  + (−1.41 + 1.41i)3-s + (−0.707 − 0.707i)7-s − 3.00i·9-s + 2.00·21-s + (2.82 + 2.82i)27-s + 2i·29-s + (1.41 + 1.41i)47-s + 1.00i·49-s + (−2.12 + 2.12i)63-s − 5.00·81-s + (1.41 − 1.41i)83-s + (−2.82 − 2.82i)87-s + (−1.41 + 1.41i)103-s + 2i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5467548628\)
\(L(\frac12)\) \(\approx\) \(0.5467548628\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
53 \( 1 - iT^{2} \)
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 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
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

−9.348360274175425999751773986259, −8.871623864588241669239116779439, −7.44830227111568611584406146175, −6.67352514138152933764771805968, −6.06998997534601949928399673173, −5.26247950388387738942071111751, −4.56400308620696911678062761602, −3.79928880784543308517575632548, −3.11518962908334704279599768356, −0.989571364621598387582665371159, 0.53262970436713733545416197201, 1.91558317804400653525364366185, 2.67013724732122697241456563560, 4.19677941262779951868702710152, 5.30592519310710094609516643930, 5.79578548620654348182970285245, 6.45559443491473804375359240989, 7.04643200751561678246016493288, 7.84945427289126939245562763675, 8.522985621972844303714302057959

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