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Properties

Label 2-2600-104.21-c0-0-0
Degree 22
Conductor 26002600
Sign −0.957−0.289i-0.957 - 0.289i
Analytic cond. 1.297561.29756
Root an. cond. 1.139101.13910
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(2600s/2 ΓC(s) L(s)=((−0.957−0.289i) Λ‾(1−s)\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}
Λ(s)=(2600s/2 ΓC(s) L(s)=((−0.957−0.289i) Λ‾(1−s)\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: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: −0.957−0.289i-0.957 - 0.289i
Analytic conductor: 1.297561.29756
Root analytic conductor: 1.139101.13910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2600(2101,⋅)\chi_{2600} (2101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2600, ( :0), −0.957−0.289i)(2,\ 2600,\ (\ :0),\ -0.957 - 0.289i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.80811325520.8081132552
L(12)L(\frac12) ≈\approx 0.80811325520.8081132552
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
5 1 1
13 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good3 1−1.41iT−T2 1 - 1.41iT - T^{2}
7 1−iT2 1 - iT^{2}
11 1+iT2 1 + iT^{2}
17 1−T2 1 - T^{2}
19 1−iT2 1 - iT^{2}
23 1−T2 1 - T^{2}
29 1−T2 1 - T^{2}
31 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
37 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
41 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
43 1+1.41T+T2 1 + 1.41T + T^{2}
47 1−iT2 1 - iT^{2}
53 1+1.41iT−T2 1 + 1.41iT - T^{2}
59 1+iT2 1 + iT^{2}
61 1−T2 1 - T^{2}
67 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
71 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
73 1−iT2 1 - iT^{2}
79 1+T2 1 + T^{2}
83 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
89 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
97 1+iT2 1 + iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line