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Properties

Label 2-15-3.2-c2-0-1
Degree $2$
Conductor $15$
Sign $0.745 + 0.666i$
Analytic cond. $0.408720$
Root an. cond. $0.639312$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23i·2-s + (−2 + 2.23i)3-s − 1.00·4-s + 2.23i·5-s + (5.00 + 4.47i)6-s − 6·7-s − 6.70i·8-s + (−1.00 − 8.94i)9-s + 5.00·10-s + 4.47i·11-s + (2.00 − 2.23i)12-s + 16·13-s + 13.4i·14-s + (−5.00 − 4.47i)15-s − 19·16-s + 4.47i·17-s + ⋯
L(s)  = 1  − 1.11i·2-s + (−0.666 + 0.745i)3-s − 0.250·4-s + 0.447i·5-s + (0.833 + 0.745i)6-s − 0.857·7-s − 0.838i·8-s + (−0.111 − 0.993i)9-s + 0.500·10-s + 0.406i·11-s + (0.166 − 0.186i)12-s + 1.23·13-s + 0.958i·14-s + (−0.333 − 0.298i)15-s − 1.18·16-s + 0.263i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.745 + 0.666i$
Analytic conductor: \(0.408720\)
Root analytic conductor: \(0.639312\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :1),\ 0.745 + 0.666i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.646624 - 0.246988i\)
\(L(\frac12)\) \(\approx\) \(0.646624 - 0.246988i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2 - 2.23i)T \)
5 \( 1 - 2.23iT \)
good2 \( 1 + 2.23iT - 4T^{2} \)
7 \( 1 + 6T + 49T^{2} \)
11 \( 1 - 4.47iT - 121T^{2} \)
13 \( 1 - 16T + 169T^{2} \)
17 \( 1 - 4.47iT - 289T^{2} \)
19 \( 1 + 2T + 361T^{2} \)
23 \( 1 - 13.4iT - 529T^{2} \)
29 \( 1 + 31.3iT - 841T^{2} \)
31 \( 1 + 18T + 961T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 - 62.6iT - 1.68e3T^{2} \)
43 \( 1 - 16T + 1.84e3T^{2} \)
47 \( 1 + 49.1iT - 2.20e3T^{2} \)
53 \( 1 - 4.47iT - 2.80e3T^{2} \)
59 \( 1 - 4.47iT - 3.48e3T^{2} \)
61 \( 1 - 82T + 3.72e3T^{2} \)
67 \( 1 - 24T + 4.48e3T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 + 74T + 5.32e3T^{2} \)
79 \( 1 - 138T + 6.24e3T^{2} \)
83 \( 1 + 93.9iT - 6.88e3T^{2} \)
89 \( 1 - 107. iT - 7.92e3T^{2} \)
97 \( 1 + 166T + 9.40e3T^{2} \)
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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

−19.26169490422754119912058216377, −18.01263456560359807346814205390, −16.33663545296476791170032337065, −15.31183624438809672746134681529, −13.14136327212728335570193545447, −11.73076231917341402814225469725, −10.63652752406966828619200485829, −9.556881881402575780202953749311, −6.39278214583289871634882090431, −3.64213518169807278952791352950, 5.68732210018541315600141812996, 6.89108609505129160058815916512, 8.555588049631488199187956317696, 11.05046670259334216137570596707, 12.68544331486549814745919762072, 13.95866256681387457806397154503, 15.92254789929129234289959207944, 16.49831964637036568070713956991, 17.76606447551716060483375990777, 19.04285858142557514784321773720

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