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Properties

Label 2-15-3.2-c4-0-0
Degree $2$
Conductor $15$
Sign $-0.461 - 0.887i$
Analytic cond. $1.55054$
Root an. cond. $1.24521$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.56i·2-s + (−7.98 + 4.15i)3-s − 4.85·4-s + 11.1i·5-s + (−18.9 − 36.4i)6-s + 61.6·7-s + 50.9i·8-s + (46.4 − 66.3i)9-s − 51.0·10-s − 108. i·11-s + (38.7 − 20.1i)12-s − 63.7·13-s + 281. i·14-s + (−46.4 − 89.2i)15-s − 310.·16-s + 175. i·17-s + ⋯
L(s)  = 1  + 1.14i·2-s + (−0.887 + 0.461i)3-s − 0.303·4-s + 0.447i·5-s + (−0.526 − 1.01i)6-s + 1.25·7-s + 0.795i·8-s + (0.573 − 0.818i)9-s − 0.510·10-s − 0.899i·11-s + (0.268 − 0.139i)12-s − 0.376·13-s + 1.43i·14-s + (−0.206 − 0.396i)15-s − 1.21·16-s + 0.606i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 - 0.887i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.461 - 0.887i$
Analytic conductor: \(1.55054\)
Root analytic conductor: \(1.24521\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :2),\ -0.461 - 0.887i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.539143 + 0.888251i\)
\(L(\frac12)\) \(\approx\) \(0.539143 + 0.888251i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.98 - 4.15i)T \)
5 \( 1 - 11.1iT \)
good2 \( 1 - 4.56iT - 16T^{2} \)
7 \( 1 - 61.6T + 2.40e3T^{2} \)
11 \( 1 + 108. iT - 1.46e4T^{2} \)
13 \( 1 + 63.7T + 2.85e4T^{2} \)
17 \( 1 - 175. iT - 8.35e4T^{2} \)
19 \( 1 - 301.T + 1.30e5T^{2} \)
23 \( 1 + 1.03e3iT - 2.79e5T^{2} \)
29 \( 1 - 179. iT - 7.07e5T^{2} \)
31 \( 1 - 1.06e3T + 9.23e5T^{2} \)
37 \( 1 + 1.06e3T + 1.87e6T^{2} \)
41 \( 1 - 173. iT - 2.82e6T^{2} \)
43 \( 1 + 3.03e3T + 3.41e6T^{2} \)
47 \( 1 + 935. iT - 4.87e6T^{2} \)
53 \( 1 - 423. iT - 7.89e6T^{2} \)
59 \( 1 + 3.18e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.30e3T + 1.38e7T^{2} \)
67 \( 1 - 4.21e3T + 2.01e7T^{2} \)
71 \( 1 - 475. iT - 2.54e7T^{2} \)
73 \( 1 + 2.14e3T + 2.83e7T^{2} \)
79 \( 1 - 3.10e3T + 3.89e7T^{2} \)
83 \( 1 - 3.76e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.26e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.31e3T + 8.85e7T^{2} \)
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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

−18.42828928849302984839721631821, −17.37716898961728438584097758589, −16.43576250342377286682723243924, −15.18683964439902794003220973724, −14.21502438377815834157569315437, −11.75910458759032937801974953305, −10.65449381650564773982663754369, −8.269509247895973920460388757882, −6.54401160512865343367343061001, −5.04728743721129814869210372854, 1.54783535168467047008664143431, 4.92795468133220458975084878699, 7.42894828303382974101147428451, 9.907941098415344876245097753995, 11.46530765173883054355818222306, 12.05232406525552351931196207144, 13.53005100117016803297618176110, 15.63378866351040225465567311674, 17.31476310240491815975141531059, 18.19868528378082897602602622062

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