Properties

Label 2-2800-140.27-c0-0-5
Degree $2$
Conductor $2800$
Sign $-0.727 + 0.685i$
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  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)7-s − 1.73i·11-s + (−1.22 − 1.22i)13-s + (−1.22 + 1.22i)17-s − 1.00·21-s + (0.707 + 0.707i)27-s i·29-s + (−1.22 − 1.22i)33-s − 1.73·39-s + (−0.707 − 0.707i)47-s + 1.00i·49-s + 1.73i·51-s + (−1.22 + 1.22i)77-s + 1.73·79-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)7-s − 1.73i·11-s + (−1.22 − 1.22i)13-s + (−1.22 + 1.22i)17-s − 1.00·21-s + (0.707 + 0.707i)27-s i·29-s + (−1.22 − 1.22i)33-s − 1.73·39-s + (−0.707 − 0.707i)47-s + 1.00i·49-s + 1.73i·51-s + (−1.22 + 1.22i)77-s + 1.73·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.727 + 0.685i$
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: \((2,\ 2800,\ (\ :0),\ -0.727 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.039733441\)
\(L(\frac12)\) \(\approx\) \(1.039733441\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
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 \( 1 - 1.73T + T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
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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

−8.457714101745182311596134380096, −8.021977937738477020945823597870, −7.32005853374415551431877449578, −6.46702021834482134418619937992, −5.83531502530942062233693456176, −4.75046478529581237819290956807, −3.62034092322816225368270861473, −2.97692268308369254140041407019, −2.06274917084108562298222236802, −0.55571625777988804915144207697, 2.12147644115869156985132922621, 2.61312002864699451883070136959, 3.71894925508174478877660590025, 4.69406246344594031177084865426, 4.96762859446513177932784349973, 6.56096352245600570858893375333, 6.86572215074744672041760449997, 7.74076859082581687566816730865, 8.982922602447847920837393861042, 9.281691803123108543652593144632

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