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Properties

Label 2-15-5.4-c7-0-5
Degree $2$
Conductor $15$
Sign $-0.217 + 0.975i$
Analytic cond. $4.68577$
Root an. cond. $2.16466$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.02i·2-s − 27i·3-s + 118.·4-s + (−272. − 60.8i)5-s − 81.6·6-s − 1.50e3i·7-s − 746. i·8-s − 729·9-s + (−183. + 824. i)10-s + 1.59e3·11-s − 3.20e3i·12-s + 956. i·13-s − 4.54e3·14-s + (−1.64e3 + 7.36e3i)15-s + 1.29e4·16-s + 3.24e4i·17-s + ⋯
L(s)  = 1  − 0.267i·2-s − 0.577i·3-s + 0.928·4-s + (−0.975 − 0.217i)5-s − 0.154·6-s − 1.65i·7-s − 0.515i·8-s − 0.333·9-s + (−0.0581 + 0.260i)10-s + 0.361·11-s − 0.536i·12-s + 0.120i·13-s − 0.443·14-s + (−0.125 + 0.563i)15-s + 0.790·16-s + 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.975i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.217 + 0.975i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(2.16466\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :7/2),\ -0.217 + 0.975i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.952915 - 1.18897i\)
\(L(\frac12)\) \(\approx\) \(0.952915 - 1.18897i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27iT \)
5 \( 1 + (272. + 60.8i)T \)
good2 \( 1 + 3.02iT - 128T^{2} \)
7 \( 1 + 1.50e3iT - 8.23e5T^{2} \)
11 \( 1 - 1.59e3T + 1.94e7T^{2} \)
13 \( 1 - 956. iT - 6.27e7T^{2} \)
17 \( 1 - 3.24e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.91e4T + 8.93e8T^{2} \)
23 \( 1 + 5.93e4iT - 3.40e9T^{2} \)
29 \( 1 + 6.61e4T + 1.72e10T^{2} \)
31 \( 1 + 1.96e4T + 2.75e10T^{2} \)
37 \( 1 - 3.76e5iT - 9.49e10T^{2} \)
41 \( 1 - 3.85e5T + 1.94e11T^{2} \)
43 \( 1 + 4.66e5iT - 2.71e11T^{2} \)
47 \( 1 - 4.68e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.60e6iT - 1.17e12T^{2} \)
59 \( 1 - 2.04e6T + 2.48e12T^{2} \)
61 \( 1 + 3.78e5T + 3.14e12T^{2} \)
67 \( 1 + 4.64e3iT - 6.06e12T^{2} \)
71 \( 1 + 2.79e6T + 9.09e12T^{2} \)
73 \( 1 - 2.01e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.76e6T + 1.92e13T^{2} \)
83 \( 1 - 3.06e6iT - 2.71e13T^{2} \)
89 \( 1 - 6.14e6T + 4.42e13T^{2} \)
97 \( 1 - 3.02e6iT - 8.07e13T^{2} \)
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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

−17.14199492346450566043192512679, −16.20728536299363999467440306414, −14.61426981307922542910250358813, −12.95229543768393663698708584536, −11.64899045157677474766211495722, −10.49881121783389078680762726301, −7.889463329237868170782291912321, −6.79796004109438481164158682349, −3.75566050579780015979135052653, −1.07340416028335586127905176930, 2.93404545943294133096175318795, 5.52301218756054743353758164113, 7.48695049899021413266679322409, 9.227818616638076575403179752492, 11.37414992854583366729582343797, 11.98776530100370150298683434185, 14.60837803121294088202819970713, 15.66107023624881542060912272482, 16.15360982175283577363944666380, 18.16052199747145976975520391415

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