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Properties

Label 2-5-5.4-c19-0-6
Degree $2$
Conductor $5$
Sign $-0.934 + 0.355i$
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  − 674. i·2-s + 3.54e4i·3-s + 6.96e4·4-s + (−4.08e6 + 1.55e6i)5-s + 2.38e7·6-s − 1.27e8i·7-s − 4.00e8i·8-s − 9.32e7·9-s + (1.04e9 + 2.75e9i)10-s − 1.50e10·11-s + 2.46e9i·12-s − 2.83e10i·13-s − 8.61e10·14-s + (−5.49e10 − 1.44e11i)15-s − 2.33e11·16-s − 2.72e10i·17-s + ⋯
L(s)  = 1  − 0.931i·2-s + 1.03i·3-s + 0.132·4-s + (−0.934 + 0.355i)5-s + 0.967·6-s − 1.19i·7-s − 1.05i·8-s − 0.0802·9-s + (0.330 + 0.870i)10-s − 1.92·11-s + 0.138i·12-s − 0.741i·13-s − 1.11·14-s + (−0.369 − 0.971i)15-s − 0.849·16-s − 0.0556i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.934 + 0.355i$
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: \((2,\ 5,\ (\ :19/2),\ -0.934 + 0.355i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.138170 - 0.752317i\)
\(L(\frac12)\) \(\approx\) \(0.138170 - 0.752317i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.08e6 - 1.55e6i)T \)
good2 \( 1 + 674. iT - 5.24e5T^{2} \)
3 \( 1 - 3.54e4iT - 1.16e9T^{2} \)
7 \( 1 + 1.27e8iT - 1.13e16T^{2} \)
11 \( 1 + 1.50e10T + 6.11e19T^{2} \)
13 \( 1 + 2.83e10iT - 1.46e21T^{2} \)
17 \( 1 + 2.72e10iT - 2.39e23T^{2} \)
19 \( 1 + 5.36e11T + 1.97e24T^{2} \)
23 \( 1 + 1.00e13iT - 7.46e25T^{2} \)
29 \( 1 + 6.42e13T + 6.10e27T^{2} \)
31 \( 1 + 4.84e12T + 2.16e28T^{2} \)
37 \( 1 - 3.59e14iT - 6.24e29T^{2} \)
41 \( 1 - 5.83e14T + 4.39e30T^{2} \)
43 \( 1 + 2.07e15iT - 1.08e31T^{2} \)
47 \( 1 - 1.09e16iT - 5.88e31T^{2} \)
53 \( 1 - 3.14e16iT - 5.77e32T^{2} \)
59 \( 1 - 6.14e16T + 4.42e33T^{2} \)
61 \( 1 - 8.81e15T + 8.34e33T^{2} \)
67 \( 1 + 2.91e17iT - 4.95e34T^{2} \)
71 \( 1 + 6.11e17T + 1.49e35T^{2} \)
73 \( 1 + 2.01e17iT - 2.53e35T^{2} \)
79 \( 1 - 7.58e17T + 1.13e36T^{2} \)
83 \( 1 + 2.69e18iT - 2.90e36T^{2} \)
89 \( 1 + 3.94e18T + 1.09e37T^{2} \)
97 \( 1 + 4.27e18iT - 5.60e37T^{2} \)
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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

−18.68091109705627838693567689773, −16.26027216730784366689210410887, −15.29117890226532274733711752358, −12.86867745763357779638533839984, −10.76911582862275869852126106198, −10.38650166360178693135977065244, −7.53084352841505807571505114273, −4.35891039511971729915563440192, −3.01561736054509416054805534549, −0.31970905131449460091552508910, 2.21183286693286771553106214346, 5.45240308834153565353557527992, 7.25883133510291488012789057423, 8.319331262554106992677345123528, 11.60248464860871962591359520190, 12.98888542352147319288794650243, 15.20357044868474339007973801112, 16.11700083882483599038410788177, 18.11386564525841509051407676977, 19.22624527973788438525048043200

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