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Properties

Label 2-5-5.4-c15-0-1
Degree $2$
Conductor $5$
Sign $-0.917 - 0.396i$
Analytic cond. $7.13467$
Root an. cond. $2.67108$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 43.0i·2-s + 6.72e3i·3-s + 3.09e4·4-s + (−1.60e5 − 6.92e4i)5-s − 2.89e5·6-s + 1.29e6i·7-s + 2.74e6i·8-s − 3.08e7·9-s + (2.98e6 − 6.91e6i)10-s − 2.48e7·11-s + 2.07e8i·12-s − 1.64e8i·13-s − 5.59e7·14-s + (4.66e8 − 1.07e9i)15-s + 8.94e8·16-s + 2.49e9i·17-s + ⋯
L(s)  = 1  + 0.238i·2-s + 1.77i·3-s + 0.943·4-s + (−0.917 − 0.396i)5-s − 0.422·6-s + 0.595i·7-s + 0.462i·8-s − 2.15·9-s + (0.0944 − 0.218i)10-s − 0.384·11-s + 1.67i·12-s − 0.728i·13-s − 0.141·14-s + (0.704 − 1.62i)15-s + 0.833·16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.917 - 0.396i$
Analytic conductor: \(7.13467\)
Root analytic conductor: \(2.67108\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :15/2),\ -0.917 - 0.396i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.296651 + 1.43456i\)
\(L(\frac12)\) \(\approx\) \(0.296651 + 1.43456i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.60e5 + 6.92e4i)T \)
good2 \( 1 - 43.0iT - 3.27e4T^{2} \)
3 \( 1 - 6.72e3iT - 1.43e7T^{2} \)
7 \( 1 - 1.29e6iT - 4.74e12T^{2} \)
11 \( 1 + 2.48e7T + 4.17e15T^{2} \)
13 \( 1 + 1.64e8iT - 5.11e16T^{2} \)
17 \( 1 - 2.49e9iT - 2.86e18T^{2} \)
19 \( 1 - 3.47e9T + 1.51e19T^{2} \)
23 \( 1 - 1.39e10iT - 2.66e20T^{2} \)
29 \( 1 - 7.33e10T + 8.62e21T^{2} \)
31 \( 1 - 6.63e9T + 2.34e22T^{2} \)
37 \( 1 - 8.47e11iT - 3.33e23T^{2} \)
41 \( 1 + 7.77e11T + 1.55e24T^{2} \)
43 \( 1 + 2.03e12iT - 3.17e24T^{2} \)
47 \( 1 + 3.83e12iT - 1.20e25T^{2} \)
53 \( 1 + 2.49e12iT - 7.31e25T^{2} \)
59 \( 1 - 1.33e12T + 3.65e26T^{2} \)
61 \( 1 - 7.16e12T + 6.02e26T^{2} \)
67 \( 1 - 6.61e12iT - 2.46e27T^{2} \)
71 \( 1 - 1.42e13T + 5.87e27T^{2} \)
73 \( 1 + 5.85e13iT - 8.90e27T^{2} \)
79 \( 1 - 2.48e14T + 2.91e28T^{2} \)
83 \( 1 + 6.66e13iT - 6.11e28T^{2} \)
89 \( 1 + 5.07e14T + 1.74e29T^{2} \)
97 \( 1 + 1.18e15iT - 6.33e29T^{2} \)
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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

−20.78042058992054161416359390144, −19.77228896634294190243840484770, −16.87654436030059573201589619943, −15.58995191319032857448406197230, −15.20718336264174933090088685410, −11.80347035941101028747301007744, −10.39165084640307282859295424913, −8.331302765718943806409977328615, −5.35987394544598620897837448375, −3.37354073923426634367942857248, 0.797286573608174255919263181731, 2.68510038089086795817311505079, 6.78719957234729993419978858596, 7.64683094838604086390406351981, 11.25598410169476265889001086924, 12.29632839791681200956070743754, 14.07780567381448724629162221098, 16.23499067037595126629729483419, 18.25622822940216093021118388508, 19.36908121378453574931584845565

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