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Properties

Label 2-15-5.4-c11-0-1
Degree $2$
Conductor $15$
Sign $0.840 - 0.541i$
Analytic cond. $11.5251$
Root an. cond. $3.39487$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 80.4i·2-s + 243i·3-s − 4.41e3·4-s + (−3.78e3 − 5.87e3i)5-s + 1.95e4·6-s + 5.39e4i·7-s + 1.90e5i·8-s − 5.90e4·9-s + (−4.72e5 + 3.04e5i)10-s + 4.07e5·11-s − 1.07e6i·12-s + 1.39e6i·13-s + 4.33e6·14-s + (1.42e6 − 9.20e5i)15-s + 6.27e6·16-s − 5.58e6i·17-s + ⋯
L(s)  = 1  − 1.77i·2-s + 0.577i·3-s − 2.15·4-s + (−0.541 − 0.840i)5-s + 1.02·6-s + 1.21i·7-s + 2.05i·8-s − 0.333·9-s + (−1.49 + 0.962i)10-s + 0.762·11-s − 1.24i·12-s + 1.03i·13-s + 2.15·14-s + (0.485 − 0.312i)15-s + 1.49·16-s − 0.953i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 0.840 - 0.541i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.555404 + 0.163534i\)
\(L(\frac12)\) \(\approx\) \(0.555404 + 0.163534i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243iT \)
5 \( 1 + (3.78e3 + 5.87e3i)T \)
good2 \( 1 + 80.4iT - 2.04e3T^{2} \)
7 \( 1 - 5.39e4iT - 1.97e9T^{2} \)
11 \( 1 - 4.07e5T + 2.85e11T^{2} \)
13 \( 1 - 1.39e6iT - 1.79e12T^{2} \)
17 \( 1 + 5.58e6iT - 3.42e13T^{2} \)
19 \( 1 + 7.86e6T + 1.16e14T^{2} \)
23 \( 1 - 6.07e7iT - 9.52e14T^{2} \)
29 \( 1 + 1.28e8T + 1.22e16T^{2} \)
31 \( 1 - 1.16e7T + 2.54e16T^{2} \)
37 \( 1 - 6.65e8iT - 1.77e17T^{2} \)
41 \( 1 + 1.25e9T + 5.50e17T^{2} \)
43 \( 1 + 8.69e8iT - 9.29e17T^{2} \)
47 \( 1 + 1.80e8iT - 2.47e18T^{2} \)
53 \( 1 + 9.98e8iT - 9.26e18T^{2} \)
59 \( 1 - 2.77e9T + 3.01e19T^{2} \)
61 \( 1 + 6.22e9T + 4.35e19T^{2} \)
67 \( 1 + 6.07e9iT - 1.22e20T^{2} \)
71 \( 1 + 3.98e9T + 2.31e20T^{2} \)
73 \( 1 + 1.70e10iT - 3.13e20T^{2} \)
79 \( 1 - 1.16e10T + 7.47e20T^{2} \)
83 \( 1 - 1.63e10iT - 1.28e21T^{2} \)
89 \( 1 + 1.50e10T + 2.77e21T^{2} \)
97 \( 1 - 1.83e10iT - 7.15e21T^{2} \)
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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

−16.95615944171236643564588398022, −15.32291876728445497142270369553, −13.52571332267237333489127046130, −11.93936667327314471566067560709, −11.55907472928820486730632449869, −9.514088337768382495537356766785, −8.837520459194425417746622058399, −5.00405129941051564072747482233, −3.60708989309448530997568908404, −1.73410737789288098590408293802, 0.26502160275698600775970147457, 4.03880979881105779259581420904, 6.27151207365398964383747297970, 7.23993800687715056544714085374, 8.329169382979872056199865625641, 10.59227504704697914739450214586, 12.91247738515877713445572773902, 14.30257882970729134738220522183, 15.01026862331234803858736929629, 16.60135678260934226948318932147

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