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Properties

Label 2-5e2-5.4-c33-0-8
Degree $2$
Conductor $25$
Sign $0.447 + 0.894i$
Analytic cond. $172.457$
Root an. cond. $13.1322$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.71e5i·2-s + 5.34e7i·3-s − 2.07e10·4-s − 9.15e12·6-s − 5.50e13i·7-s − 2.08e15i·8-s + 2.70e15·9-s − 8.18e16·11-s − 1.11e18i·12-s − 1.90e18i·13-s + 9.42e18·14-s + 1.79e20·16-s + 3.33e20i·17-s + 4.63e20i·18-s + 1.40e20·19-s + ⋯
L(s)  = 1  + 1.84i·2-s + 0.716i·3-s − 2.41·4-s − 1.32·6-s − 0.625i·7-s − 2.62i·8-s + 0.486·9-s − 0.537·11-s − 1.73i·12-s − 0.793i·13-s + 1.15·14-s + 2.43·16-s + 1.66i·17-s + 0.899i·18-s + 0.111·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(172.457\)
Root analytic conductor: \(13.1322\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :33/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(17)\) \(\approx\) \(1.011788953\)
\(L(\frac12)\) \(\approx\) \(1.011788953\)
\(L(\frac{35}{2})\) 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 \( 1 - 1.71e5iT - 8.58e9T^{2} \)
3 \( 1 - 5.34e7iT - 5.55e15T^{2} \)
7 \( 1 + 5.50e13iT - 7.73e27T^{2} \)
11 \( 1 + 8.18e16T + 2.32e34T^{2} \)
13 \( 1 + 1.90e18iT - 5.75e36T^{2} \)
17 \( 1 - 3.33e20iT - 4.02e40T^{2} \)
19 \( 1 - 1.40e20T + 1.58e42T^{2} \)
23 \( 1 - 3.12e22iT - 8.65e44T^{2} \)
29 \( 1 - 1.50e24T + 1.81e48T^{2} \)
31 \( 1 - 5.18e23T + 1.64e49T^{2} \)
37 \( 1 - 3.01e25iT - 5.63e51T^{2} \)
41 \( 1 + 2.18e26T + 1.66e53T^{2} \)
43 \( 1 - 1.76e27iT - 8.02e53T^{2} \)
47 \( 1 + 3.25e27iT - 1.51e55T^{2} \)
53 \( 1 - 9.17e27iT - 7.96e56T^{2} \)
59 \( 1 - 1.18e29T + 2.74e58T^{2} \)
61 \( 1 + 9.92e27T + 8.23e58T^{2} \)
67 \( 1 + 1.11e30iT - 1.82e60T^{2} \)
71 \( 1 - 7.58e29T + 1.23e61T^{2} \)
73 \( 1 + 6.06e30iT - 3.08e61T^{2} \)
79 \( 1 - 5.57e30T + 4.18e62T^{2} \)
83 \( 1 - 4.13e31iT - 2.13e63T^{2} \)
89 \( 1 + 6.21e31T + 2.13e64T^{2} \)
97 \( 1 + 4.04e32iT - 3.65e65T^{2} \)
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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

−12.80879492380177142704785095362, −10.53577531676005319886193181688, −9.740218941075731967716000584381, −8.379398817861071873431414492783, −7.59346725033599533235246219853, −6.44954667074274223118465155057, −5.36236872423600023931550743743, −4.43730711573230299268352256426, −3.50746022152091158879019733381, −1.13266830489321155152831898329, 0.23478862722530027689647920418, 1.06578092453934911451802601196, 2.19966231801485240598957879163, 2.72423542890499863245897275430, 4.18562675086109187871626062473, 5.18605420308471064939932757654, 6.95320191734799576983374617497, 8.464577090491302133354816451710, 9.482210652543775449272405207878, 10.50088305106941828384087421686

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