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Properties

Label 2-5-5.4-c25-0-4
Degree $2$
Conductor $5$
Sign $0.949 - 0.312i$
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  + 2.92e3i·2-s − 1.33e6i·3-s + 2.50e7·4-s + (−5.18e8 + 1.70e8i)5-s + 3.89e9·6-s + 4.60e10i·7-s + 1.71e11i·8-s − 9.29e11·9-s + (−4.98e11 − 1.51e12i)10-s + 1.55e13·11-s − 3.33e13i·12-s − 6.93e13i·13-s − 1.34e14·14-s + (2.27e14 + 6.91e14i)15-s + 3.39e14·16-s + 1.00e15i·17-s + ⋯
L(s)  = 1  + 0.504i·2-s − 1.44i·3-s + 0.745·4-s + (−0.949 + 0.312i)5-s + 0.730·6-s + 1.25i·7-s + 0.880i·8-s − 1.09·9-s + (−0.157 − 0.479i)10-s + 1.49·11-s − 1.07i·12-s − 0.824i·13-s − 0.633·14-s + (0.452 + 1.37i)15-s + 0.301·16-s + 0.419i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.949 - 0.312i$
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: \((2,\ 5,\ (\ :25/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(13)\) \(\approx\) \(2.11466 + 0.339004i\)
\(L(\frac12)\) \(\approx\) \(2.11466 + 0.339004i\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (5.18e8 - 1.70e8i)T \)
good2 \( 1 - 2.92e3iT - 3.35e7T^{2} \)
3 \( 1 + 1.33e6iT - 8.47e11T^{2} \)
7 \( 1 - 4.60e10iT - 1.34e21T^{2} \)
11 \( 1 - 1.55e13T + 1.08e26T^{2} \)
13 \( 1 + 6.93e13iT - 7.05e27T^{2} \)
17 \( 1 - 1.00e15iT - 5.77e30T^{2} \)
19 \( 1 - 9.85e15T + 9.30e31T^{2} \)
23 \( 1 - 3.90e16iT - 1.10e34T^{2} \)
29 \( 1 - 2.60e18T + 3.63e36T^{2} \)
31 \( 1 + 3.55e18T + 1.92e37T^{2} \)
37 \( 1 - 5.55e19iT - 1.60e39T^{2} \)
41 \( 1 - 1.48e20T + 2.08e40T^{2} \)
43 \( 1 + 2.65e20iT - 6.86e40T^{2} \)
47 \( 1 - 1.37e21iT - 6.34e41T^{2} \)
53 \( 1 + 1.71e21iT - 1.27e43T^{2} \)
59 \( 1 - 5.13e21T + 1.86e44T^{2} \)
61 \( 1 + 9.68e19T + 4.29e44T^{2} \)
67 \( 1 - 9.29e22iT - 4.48e45T^{2} \)
71 \( 1 + 6.03e22T + 1.91e46T^{2} \)
73 \( 1 + 3.38e22iT - 3.82e46T^{2} \)
79 \( 1 + 5.76e23T + 2.75e47T^{2} \)
83 \( 1 + 2.45e23iT - 9.48e47T^{2} \)
89 \( 1 - 1.04e24T + 5.42e48T^{2} \)
97 \( 1 + 8.72e24iT - 4.66e49T^{2} \)
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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

−17.63907665042226404561808516291, −15.77289934039713166566488176199, −14.52689869710255021983932726529, −12.30238230894726975288474125161, −11.59759205512797205877850721908, −8.354765041980793739585386295488, −7.14578735994479199196565559806, −5.99415517287751705258967629637, −2.85757430806577499020249551456, −1.27502169181343947398628548818, 0.950067761523967855472264440090, 3.55602811260975113797386123649, 4.32249421012537980159918913554, 7.07441922558983396192226779482, 9.391746696133399570797600067633, 10.81945129068792011519646142815, 11.85139692909489982386268169527, 14.45330895923298195227550326940, 16.05166982227889712819684494587, 16.65516738972443807664585783630

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