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Properties

Label 2-15-5.2-c2-0-0
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: $\chi_{15} (7, \cdot )$
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

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

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