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Properties

Label 2-5-5.4-c19-0-3
Degree $2$
Conductor $5$
Sign $0.508 - 0.860i$
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  + 113. i·2-s + 3.31e4i·3-s + 5.11e5·4-s + (2.22e6 − 3.75e6i)5-s − 3.75e6·6-s + 2.70e7i·7-s + 1.17e8i·8-s + 6.28e7·9-s + (4.26e8 + 2.51e8i)10-s + 5.56e9·11-s + 1.69e10i·12-s + 4.21e10i·13-s − 3.06e9·14-s + (1.24e11 + 7.36e10i)15-s + 2.54e11·16-s + 5.95e11i·17-s + ⋯
L(s)  = 1  + 0.156i·2-s + 0.972i·3-s + 0.975·4-s + (0.508 − 0.860i)5-s − 0.152·6-s + 0.253i·7-s + 0.309i·8-s + 0.0540·9-s + (0.134 + 0.0796i)10-s + 0.711·11-s + 0.948i·12-s + 1.10i·13-s − 0.0396·14-s + (0.837 + 0.494i)15-s + 0.927·16-s + 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.508 - 0.860i$
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: \((2,\ 5,\ (\ :19/2),\ 0.508 - 0.860i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.14847 + 1.22588i\)
\(L(\frac12)\) \(\approx\) \(2.14847 + 1.22588i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22e6 + 3.75e6i)T \)
good2 \( 1 - 113. iT - 5.24e5T^{2} \)
3 \( 1 - 3.31e4iT - 1.16e9T^{2} \)
7 \( 1 - 2.70e7iT - 1.13e16T^{2} \)
11 \( 1 - 5.56e9T + 6.11e19T^{2} \)
13 \( 1 - 4.21e10iT - 1.46e21T^{2} \)
17 \( 1 - 5.95e11iT - 2.39e23T^{2} \)
19 \( 1 + 1.45e12T + 1.97e24T^{2} \)
23 \( 1 + 9.69e12iT - 7.46e25T^{2} \)
29 \( 1 - 9.46e13T + 6.10e27T^{2} \)
31 \( 1 - 6.27e12T + 2.16e28T^{2} \)
37 \( 1 + 1.41e15iT - 6.24e29T^{2} \)
41 \( 1 + 9.33e14T + 4.39e30T^{2} \)
43 \( 1 + 2.15e13iT - 1.08e31T^{2} \)
47 \( 1 - 4.64e15iT - 5.88e31T^{2} \)
53 \( 1 + 3.42e16iT - 5.77e32T^{2} \)
59 \( 1 + 3.91e16T + 4.42e33T^{2} \)
61 \( 1 + 1.44e17T + 8.34e33T^{2} \)
67 \( 1 + 3.79e16iT - 4.95e34T^{2} \)
71 \( 1 + 7.32e17T + 1.49e35T^{2} \)
73 \( 1 + 5.16e17iT - 2.53e35T^{2} \)
79 \( 1 + 5.46e16T + 1.13e36T^{2} \)
83 \( 1 - 5.15e17iT - 2.90e36T^{2} \)
89 \( 1 - 3.08e18T + 1.09e37T^{2} \)
97 \( 1 + 5.49e18iT - 5.60e37T^{2} \)
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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

−19.54297468684518885158930567948, −17.01183659375603553243633712243, −16.12573842271066039371023646850, −14.68440647206659841343353616451, −12.34906308266013767506253045690, −10.51511123488815970337741670925, −8.860300584220159326402108815178, −6.29600916637654116772491843110, −4.32148374013859165094166361759, −1.80674809275836228471380955443, 1.31084057141007667986305372199, 2.82873058678629568874298440152, 6.33679891528075839509365594569, 7.41645435565016331695629380265, 10.30594022422330834381146193538, 11.90651158017447694471349303772, 13.57997006474467986507254831906, 15.32022085558292602741737596872, 17.35952628588402945016208379328, 18.75067640864875630943502909031

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