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Properties

Label 2-5-5.4-c25-0-8
Degree $2$
Conductor $5$
Sign $-0.596 + 0.802i$
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  − 6.31e3i·2-s − 7.11e4i·3-s − 6.31e6·4-s + (3.25e8 − 4.38e8i)5-s − 4.49e8·6-s + 3.34e10i·7-s − 1.72e11i·8-s + 8.42e11·9-s + (−2.76e12 − 2.05e12i)10-s + 1.82e13·11-s + 4.48e11i·12-s − 9.99e13i·13-s + 2.11e14·14-s + (−3.11e13 − 2.31e13i)15-s − 1.29e15·16-s + 2.20e15i·17-s + ⋯
L(s)  = 1  − 1.08i·2-s − 0.0772i·3-s − 0.188·4-s + (0.596 − 0.802i)5-s − 0.0842·6-s + 0.913i·7-s − 0.884i·8-s + 0.994·9-s + (−0.875 − 0.649i)10-s + 1.74·11-s + 0.0145i·12-s − 1.18i·13-s + 0.996·14-s + (−0.0620 − 0.0460i)15-s − 1.15·16-s + 0.916i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.596 + 0.802i$
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: \((2,\ 5,\ (\ :25/2),\ -0.596 + 0.802i)\)

Particular Values

\(L(13)\) \(\approx\) \(1.15555 - 2.29783i\)
\(L(\frac12)\) \(\approx\) \(1.15555 - 2.29783i\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.25e8 + 4.38e8i)T \)
good2 \( 1 + 6.31e3iT - 3.35e7T^{2} \)
3 \( 1 + 7.11e4iT - 8.47e11T^{2} \)
7 \( 1 - 3.34e10iT - 1.34e21T^{2} \)
11 \( 1 - 1.82e13T + 1.08e26T^{2} \)
13 \( 1 + 9.99e13iT - 7.05e27T^{2} \)
17 \( 1 - 2.20e15iT - 5.77e30T^{2} \)
19 \( 1 + 1.27e16T + 9.30e31T^{2} \)
23 \( 1 + 7.08e16iT - 1.10e34T^{2} \)
29 \( 1 + 2.12e18T + 3.63e36T^{2} \)
31 \( 1 - 4.58e18T + 1.92e37T^{2} \)
37 \( 1 - 1.67e19iT - 1.60e39T^{2} \)
41 \( 1 - 4.79e19T + 2.08e40T^{2} \)
43 \( 1 + 1.55e20iT - 6.86e40T^{2} \)
47 \( 1 + 8.97e20iT - 6.34e41T^{2} \)
53 \( 1 - 6.78e21iT - 1.27e43T^{2} \)
59 \( 1 - 8.34e21T + 1.86e44T^{2} \)
61 \( 1 + 8.24e21T + 4.29e44T^{2} \)
67 \( 1 + 8.00e21iT - 4.48e45T^{2} \)
71 \( 1 - 8.71e21T + 1.91e46T^{2} \)
73 \( 1 - 1.89e23iT - 3.82e46T^{2} \)
79 \( 1 + 1.13e23T + 2.75e47T^{2} \)
83 \( 1 + 9.87e23iT - 9.48e47T^{2} \)
89 \( 1 + 9.13e23T + 5.42e48T^{2} \)
97 \( 1 - 1.03e24iT - 4.66e49T^{2} \)
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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

−17.07494197655536627697807879739, −15.25097036623510383716758153259, −12.94552296243360503445949986084, −12.14696892611109228469488023487, −10.26979479431976885586826151140, −8.888708436221354469656528146353, −6.25530326977748927691239825035, −4.09532373881009813092154070832, −2.10208015977838690701543491387, −1.04003679870736899697763655960, 1.72352262129698525165110278559, 4.18610857545065254914799857134, 6.46992761019459936767451814851, 7.14462669369635382882122159384, 9.469445707157948239504084102911, 11.27530652856560156139040306691, 13.81664222160352109543622222571, 14.78905523544868302121216819198, 16.47446269199465489925503464556, 17.47847944658172749772803355890

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