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Properties

Label 2-2800-560.349-c0-0-0
Degree $2$
Conductor $2800$
Sign $0.655 - 0.755i$
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  − 2-s + 4-s − 7-s − 8-s − i·9-s + (−1 + i)11-s + 14-s + 16-s + i·18-s + (1 − i)22-s − 28-s + (1 + i)29-s − 32-s − i·36-s + (1 + i)37-s + ⋯
L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − i·9-s + (−1 + i)11-s + 14-s + 16-s + i·18-s + (1 − i)22-s − 28-s + (1 + i)29-s − 32-s − i·36-s + (1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.655 - 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5617900871\)
\(L(\frac12)\) \(\approx\) \(0.5617900871\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.200472253571314502173144251965, −8.430060305915447396797415858500, −7.58180308734672494215839796834, −6.88593754623943232424268765797, −6.33969862883068448902165895606, −5.47039880041512502445040689723, −4.26107348719692623995583089648, −3.11353231397606695342027140404, −2.51598418070910656439914318170, −1.04777783004992994368628790204, 0.58019927509636180084218594227, 2.28995185911227562548074960459, 2.83060586188879965332347039221, 3.92505069827376458488398186427, 5.31736595214568411602469463324, 5.92352991805528178600974255533, 6.72733974255722542258131088129, 7.61365625356703023209743777582, 8.116612011059271765159953584554, 8.851910939477824819076561951213

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