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Properties

Label 2-15-5.3-c2-0-1
Degree $2$
Conductor $15$
Sign $0.991 + 0.130i$
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  + (0.224 − 0.224i)2-s + (−1.22 − 1.22i)3-s + 3.89i·4-s + (−4.67 − 1.77i)5-s − 0.550·6-s + (3.44 − 3.44i)7-s + (1.77 + 1.77i)8-s + 2.99i·9-s + (−1.44 + 0.651i)10-s + 11.3·11-s + (4.77 − 4.77i)12-s + (−5.55 − 5.55i)13-s − 1.55i·14-s + (3.55 + 7.89i)15-s − 14.7·16-s + (−17.3 + 17.3i)17-s + ⋯
L(s)  = 1  + (0.112 − 0.112i)2-s + (−0.408 − 0.408i)3-s + 0.974i·4-s + (−0.934 − 0.355i)5-s − 0.0917·6-s + (0.492 − 0.492i)7-s + (0.221 + 0.221i)8-s + 0.333i·9-s + (−0.144 + 0.0651i)10-s + 1.03·11-s + (0.397 − 0.397i)12-s + (−0.426 − 0.426i)13-s − 0.110i·14-s + (0.236 + 0.526i)15-s − 0.924·16-s + (−1.02 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(0.408720\)
Root analytic conductor: \(0.639312\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :1),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.712668 - 0.0467985i\)
\(L(\frac12)\) \(\approx\) \(0.712668 - 0.0467985i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (4.67 + 1.77i)T \)
good2 \( 1 + (-0.224 + 0.224i)T - 4iT^{2} \)
7 \( 1 + (-3.44 + 3.44i)T - 49iT^{2} \)
11 \( 1 - 11.3T + 121T^{2} \)
13 \( 1 + (5.55 + 5.55i)T + 169iT^{2} \)
17 \( 1 + (17.3 - 17.3i)T - 289iT^{2} \)
19 \( 1 + 8.69iT - 361T^{2} \)
23 \( 1 + (-11.5 - 11.5i)T + 529iT^{2} \)
29 \( 1 + 35.1iT - 841T^{2} \)
31 \( 1 - 10.6T + 961T^{2} \)
37 \( 1 + (6.04 - 6.04i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.696T + 1.68e3T^{2} \)
43 \( 1 + (26.4 + 26.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-44.2 + 44.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (0.696 + 0.696i)T + 2.80e3iT^{2} \)
59 \( 1 - 39.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.90T + 3.72e3T^{2} \)
67 \( 1 + (45.1 - 45.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-13.1 - 13.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.5 - 24.5i)T - 9.40e3iT^{2} \)
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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

−19.38545099124929905717538849349, −17.50501961280439289905304112007, −16.91880099812900574531630198919, −15.31558856257637995376901438599, −13.40415428810468843990665511310, −12.15455676244450934468808544599, −11.19155019164708753230762264607, −8.502508879335912284238454206158, −7.17061287013729177032608512564, −4.23786525700189323868467818503, 4.70631905387741320263632042213, 6.73548236126271204075145647014, 9.138486862222952065413425180733, 10.89979022089326742614525957579, 11.91071577448152424986163173134, 14.30687367756745500554832403620, 15.16944591749910330032757698124, 16.34205360435462487295195706961, 18.10520000283866098205353209864, 19.24650811860761799651666020340

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