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Properties

Label 2-5-5.4-c25-0-3
Degree $2$
Conductor $5$
Sign $-0.770 + 0.637i$
Analytic cond. $19.7998$
Root an. cond. $4.44970$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.68e3i·2-s + 1.63e6i·3-s + 1.27e6·4-s + (4.20e8 − 3.47e8i)5-s − 9.31e9·6-s + 6.01e10i·7-s + 1.97e11i·8-s − 1.84e12·9-s + (1.97e12 + 2.38e12i)10-s + 1.89e11·11-s + 2.09e12i·12-s − 1.09e13i·13-s − 3.41e14·14-s + (5.70e14 + 6.89e14i)15-s − 1.08e15·16-s − 1.44e15i·17-s + ⋯
L(s)  = 1  + 0.980i·2-s + 1.78i·3-s + 0.0381·4-s + (0.770 − 0.637i)5-s − 1.74·6-s + 1.64i·7-s + 1.01i·8-s − 2.17·9-s + (0.625 + 0.755i)10-s + 0.0182·11-s + 0.0678i·12-s − 0.130i·13-s − 1.61·14-s + (1.13 + 1.37i)15-s − 0.960·16-s − 0.602i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.770 + 0.637i$
Analytic conductor: \(19.7998\)
Root analytic conductor: \(4.44970\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :25/2),\ -0.770 + 0.637i)\)

Particular Values

\(L(13)\) \(\approx\) \(0.758357 - 2.10632i\)
\(L(\frac12)\) \(\approx\) \(0.758357 - 2.10632i\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.20e8 + 3.47e8i)T \)
good2 \( 1 - 5.68e3iT - 3.35e7T^{2} \)
3 \( 1 - 1.63e6iT - 8.47e11T^{2} \)
7 \( 1 - 6.01e10iT - 1.34e21T^{2} \)
11 \( 1 - 1.89e11T + 1.08e26T^{2} \)
13 \( 1 + 1.09e13iT - 7.05e27T^{2} \)
17 \( 1 + 1.44e15iT - 5.77e30T^{2} \)
19 \( 1 - 1.43e16T + 9.30e31T^{2} \)
23 \( 1 - 1.47e16iT - 1.10e34T^{2} \)
29 \( 1 - 1.47e16T + 3.63e36T^{2} \)
31 \( 1 - 3.62e18T + 1.92e37T^{2} \)
37 \( 1 + 4.70e19iT - 1.60e39T^{2} \)
41 \( 1 + 5.52e19T + 2.08e40T^{2} \)
43 \( 1 - 5.36e19iT - 6.86e40T^{2} \)
47 \( 1 - 3.09e20iT - 6.34e41T^{2} \)
53 \( 1 + 1.46e21iT - 1.27e43T^{2} \)
59 \( 1 + 1.97e21T + 1.86e44T^{2} \)
61 \( 1 - 2.98e22T + 4.29e44T^{2} \)
67 \( 1 - 5.17e22iT - 4.48e45T^{2} \)
71 \( 1 + 9.98e22T + 1.91e46T^{2} \)
73 \( 1 - 2.29e23iT - 3.82e46T^{2} \)
79 \( 1 + 2.98e23T + 2.75e47T^{2} \)
83 \( 1 - 5.05e23iT - 9.48e47T^{2} \)
89 \( 1 + 9.04e23T + 5.42e48T^{2} \)
97 \( 1 - 7.24e24iT - 4.66e49T^{2} \)
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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

−17.69352416921622069980301260556, −16.22848765280818288915090127714, −15.63095684976484021198938784160, −14.36297279620572184056948670850, −11.64858685050082188571823913012, −9.647792509669402418200903028657, −8.638820998991617534312510476853, −5.73499060102968981452662622719, −5.08319660509057306243049268847, −2.65382240312968012217343570406, 0.830617792918150923400017378704, 1.69241742937725165946569695135, 3.14079714901499528333767722918, 6.49313991931778908588567274763, 7.42519390325392340402869026852, 10.23169669413976080988006945054, 11.60799757832723849149191201237, 13.17292492891610125358344008013, 13.95768757596317419182181676884, 17.12230716849661457002732505023

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