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Properties

Label 2-2600-13.8-c0-0-1
Degree $2$
Conductor $2600$
Sign $0.471 + 0.881i$
Analytic cond. $1.29756$
Root an. cond. $1.13910$
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  − 3-s + (1 − i)7-s + (−1 + i)11-s + 13-s − 2i·17-s + (−1 + i)21-s + i·23-s + 27-s + 29-s + (−1 − i)31-s + (1 − i)33-s − 39-s − i·43-s − i·49-s + 2i·51-s + ⋯
L(s)  = 1  − 3-s + (1 − i)7-s + (−1 + i)11-s + 13-s − 2i·17-s + (−1 + i)21-s + i·23-s + 27-s + 29-s + (−1 − i)31-s + (1 − i)33-s − 39-s − i·43-s − i·49-s + 2i·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.471 + 0.881i$
Analytic conductor: \(1.29756\)
Root analytic conductor: \(1.13910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2600,\ (\ :0),\ 0.471 + 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8348378001\)
\(L(\frac12)\) \(\approx\) \(0.8348378001\)
\(L(1)\) 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 \( 1 \)
13 \( 1 - T \)
good3 \( 1 + T + T^{2} \)
7 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1 + i)T - iT^{2} \)
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

−8.974335055000032298610816652900, −7.88067564767536930508358346433, −7.45443314238938240696986919500, −6.70372248788775191879185165806, −5.65510493486234228790425747035, −4.99559791657977501237391734408, −4.50403868623672953307677643369, −3.28875099250166150528665345959, −1.99770493597723455456818382992, −0.69893219506802027098242103427, 1.26884347162068186505587203087, 2.47430384534340630315478143657, 3.57220742515065482532847037175, 4.75783534139078831867534267904, 5.41791615040720902154346114464, 6.03051350823542971735633980737, 6.50768722269845412283754053266, 8.004314595349097535579198092902, 8.421238248567540161118263750515, 8.828088370012134247305171807952

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