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Properties

Label 2-5e2-5.2-c32-0-29
Degree $2$
Conductor $25$
Sign $0.850 + 0.525i$
Analytic cond. $162.166$
Root an. cond. $12.7344$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.09e3 + 7.09e3i)2-s + (4.70e7 − 4.70e7i)3-s − 4.19e9i·4-s + 6.67e11·6-s + (3.04e13 + 3.04e13i)7-s + (6.02e13 − 6.02e13i)8-s − 2.57e15i·9-s + 3.11e16·11-s + (−1.97e17 − 1.97e17i)12-s + (−8.11e17 + 8.11e17i)13-s + 4.32e17i·14-s − 1.71e19·16-s + (5.99e19 + 5.99e19i)17-s + (1.82e19 − 1.82e19i)18-s + 3.24e19i·19-s + ⋯
L(s)  = 1  + (0.108 + 0.108i)2-s + (1.09 − 1.09i)3-s − 0.976i·4-s + 0.236·6-s + (0.915 + 0.915i)7-s + (0.214 − 0.214i)8-s − 1.38i·9-s + 0.678·11-s + (−1.06 − 1.06i)12-s + (−1.21 + 1.21i)13-s + 0.198i·14-s − 0.930·16-s + (1.23 + 1.23i)17-s + (0.150 − 0.150i)18-s + 0.112i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(162.166\)
Root analytic conductor: \(12.7344\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :16),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(4.523639836\)
\(L(\frac12)\) \(\approx\) \(4.523639836\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-7.09e3 - 7.09e3i)T + 4.29e9iT^{2} \)
3 \( 1 + (-4.70e7 + 4.70e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (-3.04e13 - 3.04e13i)T + 1.10e27iT^{2} \)
11 \( 1 - 3.11e16T + 2.11e33T^{2} \)
13 \( 1 + (8.11e17 - 8.11e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (-5.99e19 - 5.99e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 3.24e19iT - 8.31e40T^{2} \)
23 \( 1 + (7.07e20 - 7.07e20i)T - 3.76e43iT^{2} \)
29 \( 1 + 1.29e23iT - 6.26e46T^{2} \)
31 \( 1 - 1.94e23T + 5.29e47T^{2} \)
37 \( 1 + (-1.30e25 - 1.30e25i)T + 1.52e50iT^{2} \)
41 \( 1 - 1.97e25T + 4.06e51T^{2} \)
43 \( 1 + (-1.49e26 + 1.49e26i)T - 1.86e52iT^{2} \)
47 \( 1 + (-1.46e25 - 1.46e25i)T + 3.21e53iT^{2} \)
53 \( 1 + (1.61e26 - 1.61e26i)T - 1.50e55iT^{2} \)
59 \( 1 - 1.65e28iT - 4.64e56T^{2} \)
61 \( 1 - 3.79e28T + 1.35e57T^{2} \)
67 \( 1 + (-5.62e27 - 5.62e27i)T + 2.71e58iT^{2} \)
71 \( 1 - 2.56e29T + 1.73e59T^{2} \)
73 \( 1 + (2.59e28 - 2.59e28i)T - 4.22e59iT^{2} \)
79 \( 1 - 6.46e29iT - 5.29e60T^{2} \)
83 \( 1 + (5.85e30 - 5.85e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 1.03e31iT - 2.40e62T^{2} \)
97 \( 1 + (4.86e31 + 4.86e31i)T + 3.77e63iT^{2} \)
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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

−11.59304952543980757554306719774, −9.834785873523762672551429020761, −8.843063218465839164383630675604, −7.82329187797349935042191583035, −6.69584755789916518120041484968, −5.58248931492947542403484078393, −4.22029247769199125050464646007, −2.47563708145061588393159467741, −1.79378991630431098194944443260, −1.07460195854275403286245847499, 0.76982604921924738799098513436, 2.47930767495855747705109499848, 3.26281441656329829295388385802, 4.22318491472608265940872386894, 5.01640093882210327531661190717, 7.46094815321237718052407637005, 7.918500229632975811939610405717, 9.207863277222033303648990777966, 10.16895781502719432678870052970, 11.40987656067954468100551640759

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