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Properties

Label 2-5-5.3-c16-0-3
Degree $2$
Conductor $5$
Sign $0.933 + 0.358i$
Analytic cond. $8.11622$
Root an. cond. $2.84889$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (88.8 − 88.8i)2-s + (−1.52e3 − 1.52e3i)3-s + 4.97e4i·4-s + (3.83e5 − 7.27e4i)5-s − 2.71e5·6-s + (2.52e6 − 2.52e6i)7-s + (1.02e7 + 1.02e7i)8-s − 3.83e7i·9-s + (2.76e7 − 4.05e7i)10-s + 3.78e8·11-s + (7.59e7 − 7.59e7i)12-s + (2.20e8 + 2.20e8i)13-s − 4.47e8i·14-s + (−6.97e8 − 4.75e8i)15-s − 1.44e9·16-s + (−6.87e9 + 6.87e9i)17-s + ⋯
L(s)  = 1  + (0.346 − 0.346i)2-s + (−0.232 − 0.232i)3-s + 0.759i·4-s + (0.982 − 0.186i)5-s − 0.161·6-s + (0.437 − 0.437i)7-s + (0.610 + 0.610i)8-s − 0.891i·9-s + (0.276 − 0.405i)10-s + 1.76·11-s + (0.176 − 0.176i)12-s + (0.270 + 0.270i)13-s − 0.303i·14-s + (−0.271 − 0.185i)15-s − 0.335·16-s + (−0.986 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.933 + 0.358i$
Analytic conductor: \(8.11622\)
Root analytic conductor: \(2.84889\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :8),\ 0.933 + 0.358i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(2.31622 - 0.429106i\)
\(L(\frac12)\) \(\approx\) \(2.31622 - 0.429106i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.83e5 + 7.27e4i)T \)
good2 \( 1 + (-88.8 + 88.8i)T - 6.55e4iT^{2} \)
3 \( 1 + (1.52e3 + 1.52e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (-2.52e6 + 2.52e6i)T - 3.32e13iT^{2} \)
11 \( 1 - 3.78e8T + 4.59e16T^{2} \)
13 \( 1 + (-2.20e8 - 2.20e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (6.87e9 - 6.87e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 4.03e9iT - 2.88e20T^{2} \)
23 \( 1 + (-3.57e9 - 3.57e9i)T + 6.13e21iT^{2} \)
29 \( 1 + 2.32e11iT - 2.50e23T^{2} \)
31 \( 1 + 7.26e11T + 7.27e23T^{2} \)
37 \( 1 + (-1.61e12 + 1.61e12i)T - 1.23e25iT^{2} \)
41 \( 1 - 5.73e12T + 6.37e25T^{2} \)
43 \( 1 + (1.45e13 + 1.45e13i)T + 1.36e26iT^{2} \)
47 \( 1 + (2.06e13 - 2.06e13i)T - 5.66e26iT^{2} \)
53 \( 1 + (6.41e13 + 6.41e13i)T + 3.87e27iT^{2} \)
59 \( 1 - 6.52e13iT - 2.15e28T^{2} \)
61 \( 1 + 2.25e13T + 3.67e28T^{2} \)
67 \( 1 + (4.18e14 - 4.18e14i)T - 1.64e29iT^{2} \)
71 \( 1 - 4.56e14T + 4.16e29T^{2} \)
73 \( 1 + (-8.85e14 - 8.85e14i)T + 6.50e29iT^{2} \)
79 \( 1 - 7.16e14iT - 2.30e30T^{2} \)
83 \( 1 + (4.08e14 + 4.08e14i)T + 5.07e30iT^{2} \)
89 \( 1 + 5.94e15iT - 1.54e31T^{2} \)
97 \( 1 + (-3.26e15 + 3.26e15i)T - 6.14e31iT^{2} \)
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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

−20.07878649099931947141761503685, −17.68168105872181514428136218825, −16.99431592722967443535073320413, −14.33232036135703940412587027572, −12.87459159626569518270912371768, −11.39380015630547335520377518822, −9.009083891085433092328526825934, −6.54872402964061452872887943412, −4.02217683579096243043307409786, −1.58321768430100799345468796020, 1.66740665256179102188107741316, 4.94835694151311107726804212224, 6.43657856251186295971733159547, 9.407165426405647538469375776612, 11.09488196560328233521867907901, 13.63787752712551767343787174284, 14.76532108685123946714013096759, 16.55163000914648678278849461870, 18.20606903669426749842754507778, 19.86280851980613225883160401719

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