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Properties

Label 2-5-5.4-c19-0-7
Degree $2$
Conductor $5$
Sign $-0.486 - 0.873i$
Analytic cond. $11.4408$
Root an. cond. $3.38243$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 980. i·2-s − 4.20e4i·3-s − 4.37e5·4-s + (−2.12e6 − 3.81e6i)5-s − 4.12e7·6-s − 4.16e7i·7-s − 8.52e7i·8-s − 6.07e8·9-s + (−3.74e9 + 2.08e9i)10-s + 1.14e10·11-s + 1.83e10i·12-s + 4.17e10i·13-s − 4.08e10·14-s + (−1.60e11 + 8.93e10i)15-s − 3.12e11·16-s − 7.42e9i·17-s + ⋯
L(s)  = 1  − 1.35i·2-s − 1.23i·3-s − 0.834·4-s + (−0.486 − 0.873i)5-s − 1.67·6-s − 0.389i·7-s − 0.224i·8-s − 0.522·9-s + (−1.18 + 0.658i)10-s + 1.46·11-s + 1.02i·12-s + 1.09i·13-s − 0.528·14-s + (−1.07 + 0.600i)15-s − 1.13·16-s − 0.0151i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(11.4408\)
Root analytic conductor: \(3.38243\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :19/2),\ -0.486 - 0.873i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.811937 + 1.38103i\)
\(L(\frac12)\) \(\approx\) \(0.811937 + 1.38103i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.12e6 + 3.81e6i)T \)
good2 \( 1 + 980. iT - 5.24e5T^{2} \)
3 \( 1 + 4.20e4iT - 1.16e9T^{2} \)
7 \( 1 + 4.16e7iT - 1.13e16T^{2} \)
11 \( 1 - 1.14e10T + 6.11e19T^{2} \)
13 \( 1 - 4.17e10iT - 1.46e21T^{2} \)
17 \( 1 + 7.42e9iT - 2.39e23T^{2} \)
19 \( 1 - 2.13e12T + 1.97e24T^{2} \)
23 \( 1 + 9.51e12iT - 7.46e25T^{2} \)
29 \( 1 + 5.19e13T + 6.10e27T^{2} \)
31 \( 1 + 2.16e14T + 2.16e28T^{2} \)
37 \( 1 - 7.79e14iT - 6.24e29T^{2} \)
41 \( 1 - 1.01e15T + 4.39e30T^{2} \)
43 \( 1 + 3.59e15iT - 1.08e31T^{2} \)
47 \( 1 + 5.35e14iT - 5.88e31T^{2} \)
53 \( 1 - 1.44e16iT - 5.77e32T^{2} \)
59 \( 1 - 1.09e17T + 4.42e33T^{2} \)
61 \( 1 + 1.12e17T + 8.34e33T^{2} \)
67 \( 1 - 5.23e16iT - 4.95e34T^{2} \)
71 \( 1 - 2.62e17T + 1.49e35T^{2} \)
73 \( 1 + 4.11e17iT - 2.53e35T^{2} \)
79 \( 1 - 6.87e17T + 1.13e36T^{2} \)
83 \( 1 - 1.74e18iT - 2.90e36T^{2} \)
89 \( 1 + 4.41e17T + 1.09e37T^{2} \)
97 \( 1 + 8.34e18iT - 5.60e37T^{2} \)
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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

−18.52214884704932688327579981167, −16.63921174625562846936507669615, −13.76398795118287928331622951895, −12.38373518719507068119067389040, −11.58761331368644724761862556041, −9.204449692492887514742485098509, −7.04473472253404703331352963041, −3.98130355863727855174632213388, −1.69129196686179634757714001565, −0.78821774480713855965924585185, 3.62906517344969903661762936904, 5.57642900716856323219850371192, 7.39573164421913238834310301911, 9.365915205063801798425055196222, 11.29607486473740556204045362369, 14.45043091182218774186611200934, 15.30389035058108066650282889908, 16.30026309187555320366628805548, 17.89186585928418814256013491476, 20.01982005656872311992187391774

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