Properties

Label 2-2800-140.27-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(2.103240523\)
\(L(\frac12)\) \(\approx\) \(2.103240523\)
\(L(1)\) 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

−8.605125649612419199256138984238, −8.215864155789778300892139728663, −7.36104017750431592073649656041, −6.83298328197494639637809793343, −5.96841043796301545301530426025, −4.97635516499208739163402068639, −3.69413216050077361582942058846, −2.93570284122435744531876797557, −2.05932292767583353403717760591, −1.32659965062465926214105090107, 1.79633580454938300810023398356, 2.75602755166100783356132578982, 3.63778790001779422518305702282, 4.38748499098086923225763023360, 4.81915726906227191709807959980, 5.89936702002180636902563646425, 7.24933569145496850988242972651, 7.952545329696382454953891120291, 8.338388723493983064326494027740, 9.210943462621559055696552615624

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