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Properties

Label 2-2600-104.21-c0-0-0
Degree $2$
Conductor $2600$
Sign $-0.957 - 0.289i$
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  + (−0.707 + 0.707i)2-s + 1.41i·3-s − 1.00i·4-s + (−1.00 − 1.00i)6-s + (0.707 + 0.707i)8-s − 1.00·9-s + 1.41·12-s + (0.707 + 0.707i)13-s − 1.00·16-s + (0.707 − 0.707i)18-s + (−1.00 + 1.00i)24-s − 1.00·26-s + (1 + i)31-s + (0.707 − 0.707i)32-s + 1.00i·36-s + (1.41 + 1.41i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + 1.41i·3-s − 1.00i·4-s + (−1.00 − 1.00i)6-s + (0.707 + 0.707i)8-s − 1.00·9-s + 1.41·12-s + (0.707 + 0.707i)13-s − 1.00·16-s + (0.707 − 0.707i)18-s + (−1.00 + 1.00i)24-s − 1.00·26-s + (1 + i)31-s + (0.707 − 0.707i)32-s + 1.00i·36-s + (1.41 + 1.41i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(1.29756\)
Root analytic conductor: \(1.13910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (2101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2600,\ (\ :0),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8081132552\)
\(L(\frac12)\) \(\approx\) \(0.8081132552\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
71 \( 1 + (-1 - i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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

−9.448262207019616583257270026350, −8.598613375071391152181777365912, −8.280709581630863362818292809551, −7.03369391428662270162544984966, −6.41831010691959772226522336223, −5.49032136559016528369452092738, −4.75664999266650923306729299451, −4.08937309108936352855505259778, −2.98019964983552489992020744141, −1.47596825437470975825682453908, 0.72675668397357298059262486506, 1.70982776978356270762040202918, 2.61195268713037542096315013309, 3.50805086077880500059344551321, 4.63774077337972502339645265956, 5.98038522460447283886300405691, 6.55496829176566930425471346241, 7.50563789605418818848565299641, 7.943724724931747242559699836330, 8.552406769598775436876783791474

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