Properties

Label 2-15-5.4-c3-0-0
Degree $2$
Conductor $15$
Sign $0.116 - 0.993i$
Analytic cond. $0.885028$
Root an. cond. $0.940759$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.70i·2-s − 3i·3-s − 14.1·4-s + (11.1 + 1.29i)5-s + 14.1·6-s − 16.2i·7-s − 28.7i·8-s − 9·9-s + (−6.10 + 52.2i)10-s − 40.2·11-s + 42.3i·12-s − 19.7i·13-s + 76.2·14-s + (3.89 − 33.3i)15-s + 22.1·16-s + 83.0i·17-s + ⋯
L(s)  = 1  + 1.66i·2-s − 0.577i·3-s − 1.76·4-s + (0.993 + 0.116i)5-s + 0.959·6-s − 0.875i·7-s − 1.26i·8-s − 0.333·9-s + (−0.193 + 1.65i)10-s − 1.10·11-s + 1.01i·12-s − 0.422i·13-s + 1.45·14-s + (0.0670 − 0.573i)15-s + 0.345·16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.116 - 0.993i$
Analytic conductor: \(0.885028\)
Root analytic conductor: \(0.940759\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :3/2),\ 0.116 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.746621 + 0.664408i\)
\(L(\frac12)\) \(\approx\) \(0.746621 + 0.664408i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (-11.1 - 1.29i)T \)
good2 \( 1 - 4.70iT - 8T^{2} \)
7 \( 1 + 16.2iT - 343T^{2} \)
11 \( 1 + 40.2T + 1.33e3T^{2} \)
13 \( 1 + 19.7iT - 2.19e3T^{2} \)
17 \( 1 - 83.0iT - 4.91e3T^{2} \)
19 \( 1 - 48.8T + 6.85e3T^{2} \)
23 \( 1 + 1.61iT - 1.21e4T^{2} \)
29 \( 1 - 24.5T + 2.43e4T^{2} \)
31 \( 1 + 12.4T + 2.97e4T^{2} \)
37 \( 1 - 325. iT - 5.06e4T^{2} \)
41 \( 1 + 242.T + 6.89e4T^{2} \)
43 \( 1 + 367. iT - 7.95e4T^{2} \)
47 \( 1 + 204. iT - 1.03e5T^{2} \)
53 \( 1 - 61.5iT - 1.48e5T^{2} \)
59 \( 1 - 112.T + 2.05e5T^{2} \)
61 \( 1 - 477.T + 2.26e5T^{2} \)
67 \( 1 - 558. iT - 3.00e5T^{2} \)
71 \( 1 - 558.T + 3.57e5T^{2} \)
73 \( 1 + 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3iT - 5.71e5T^{2} \)
89 \( 1 + 96.9T + 7.04e5T^{2} \)
97 \( 1 + 1.15e3iT - 9.12e5T^{2} \)
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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

−18.55030201219094461736708412328, −17.56068102147823746548863594254, −16.76287278212089790181203312457, −15.23183787881540751954472893735, −13.88675440996491493311631212870, −13.12900663952029415428842181467, −10.26743998305704963731623170871, −8.271522335076912586533471794734, −6.91590467810850540071134234992, −5.48140434504638307387215287887, 2.60559007101968636034862575648, 5.17545287819315300426887780869, 9.082106191258618449472323379130, 10.04009475161822579189447153630, 11.40902190700606852083490875718, 12.80695705481632118945539740878, 14.06037495104659114004268631134, 15.99103383261612963082722083711, 17.90150490290692913911282257516, 18.68865331108323395418444030636

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