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Properties

Label 2-15-5.4-c3-0-1
Degree $2$
Conductor $15$
Sign $0.688 - 0.724i$
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  + 1.70i·2-s + 3i·3-s + 5.10·4-s + (−8.10 − 7.70i)5-s − 5.10·6-s − 22.2i·7-s + 22.2i·8-s − 9·9-s + (13.1 − 13.7i)10-s − 1.79·11-s + 15.3i·12-s + 58.2i·13-s + 37.7·14-s + (23.1 − 24.3i)15-s + 2.89·16-s − 18.9i·17-s + ⋯
L(s)  = 1  + 0.601i·2-s + 0.577i·3-s + 0.638·4-s + (−0.724 − 0.688i)5-s − 0.347·6-s − 1.19i·7-s + 0.985i·8-s − 0.333·9-s + (0.414 − 0.436i)10-s − 0.0490·11-s + 0.368i·12-s + 1.24i·13-s + 0.721·14-s + (0.397 − 0.418i)15-s + 0.0452·16-s − 0.270i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.688 - 0.724i$
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: \((2,\ 15,\ (\ :3/2),\ 0.688 - 0.724i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.940177 + 0.403552i\)
\(L(\frac12)\) \(\approx\) \(0.940177 + 0.403552i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.10 + 7.70i)T \)
good2 \( 1 - 1.70iT - 8T^{2} \)
7 \( 1 + 22.2iT - 343T^{2} \)
11 \( 1 + 1.79T + 1.33e3T^{2} \)
13 \( 1 - 58.2iT - 2.19e3T^{2} \)
17 \( 1 + 18.9iT - 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 + 49.6iT - 1.21e4T^{2} \)
29 \( 1 - 293.T + 2.43e4T^{2} \)
31 \( 1 - 64.4T + 2.97e4T^{2} \)
37 \( 1 - 19.8iT - 5.06e4T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 + 247. iT - 7.95e4T^{2} \)
47 \( 1 + 384. iT - 1.03e5T^{2} \)
53 \( 1 - 463. iT - 1.48e5T^{2} \)
59 \( 1 - 73.7T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 173. iT - 3.00e5T^{2} \)
71 \( 1 + 594.T + 3.57e5T^{2} \)
73 \( 1 - 320. iT - 3.89e5T^{2} \)
79 \( 1 - 770.T + 4.93e5T^{2} \)
83 \( 1 - 173. iT - 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 384. iT - 9.12e5T^{2} \)
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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

−19.42381086195133821834902176725, −17.05042483898276986362979849657, −16.48025858588084541219955233814, −15.37221282991038364890764394428, −13.97056013694459173859190634953, −11.93352692971075732567821740168, −10.58724244716950306665003513674, −8.476519061928771228668739641514, −6.86705334043097671359123212326, −4.43486437003440839041916273830, 2.79345628644283790574684224363, 6.37652257153146061052060194443, 8.125903885799567260777787983999, 10.51178912745310602829285786506, 11.79519540073729265522515856540, 12.72268697022107761520899556525, 14.92608534778043455477058147175, 15.78216246636905466739539400184, 17.84825338303029670933596598148, 19.05854589981619490520107685956

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