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Properties

Label 2-15-15.8-c9-0-15
Degree $2$
Conductor $15$
Sign $-0.986 + 0.161i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.35 + 6.35i)2-s + (0.342 − 140. i)3-s − 431. i·4-s + (−1.05e3 + 919. i)5-s + (893. − 889. i)6-s + (−7.86e3 + 7.86e3i)7-s + (5.99e3 − 5.99e3i)8-s + (−1.96e4 − 96.0i)9-s + (−1.25e4 − 839. i)10-s − 5.07e4i·11-s + (−6.04e4 − 147. i)12-s + (−4.94e3 − 4.94e3i)13-s − 9.99e4·14-s + (1.28e5 + 1.47e5i)15-s − 1.44e5·16-s + (−1.74e5 − 1.74e5i)17-s + ⋯
L(s)  = 1  + (0.280 + 0.280i)2-s + (0.00243 − 0.999i)3-s − 0.842i·4-s + (−0.752 + 0.658i)5-s + (0.281 − 0.280i)6-s + (−1.23 + 1.23i)7-s + (0.517 − 0.517i)8-s + (−0.999 − 0.00487i)9-s + (−0.396 − 0.0265i)10-s − 1.04i·11-s + (−0.842 − 0.00205i)12-s + (−0.0480 − 0.0480i)13-s − 0.695·14-s + (0.656 + 0.754i)15-s − 0.551·16-s + (−0.507 − 0.507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.986 + 0.161i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.986 + 0.161i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0436933 - 0.536688i\)
\(L(\frac12)\) \(\approx\) \(0.0436933 - 0.536688i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.342 + 140. i)T \)
5 \( 1 + (1.05e3 - 919. i)T \)
good2 \( 1 + (-6.35 - 6.35i)T + 512iT^{2} \)
7 \( 1 + (7.86e3 - 7.86e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 5.07e4iT - 2.35e9T^{2} \)
13 \( 1 + (4.94e3 + 4.94e3i)T + 1.06e10iT^{2} \)
17 \( 1 + (1.74e5 + 1.74e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 3.60e4iT - 3.22e11T^{2} \)
23 \( 1 + (-9.95e5 + 9.95e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 1.37e6T + 1.45e13T^{2} \)
31 \( 1 + 5.36e6T + 2.64e13T^{2} \)
37 \( 1 + (4.18e6 - 4.18e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 1.69e7iT - 3.27e14T^{2} \)
43 \( 1 + (-4.04e6 - 4.04e6i)T + 5.02e14iT^{2} \)
47 \( 1 + (-3.93e7 - 3.93e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-4.34e7 + 4.34e7i)T - 3.29e15iT^{2} \)
59 \( 1 + 1.26e8T + 8.66e15T^{2} \)
61 \( 1 + 1.56e8T + 1.16e16T^{2} \)
67 \( 1 + (7.00e7 - 7.00e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 3.78e7iT - 4.58e16T^{2} \)
73 \( 1 + (-1.07e8 - 1.07e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 6.60e7iT - 1.19e17T^{2} \)
83 \( 1 + (3.05e8 - 3.05e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 4.49e7T + 3.50e17T^{2} \)
97 \( 1 + (1.77e8 - 1.77e8i)T - 7.60e17iT^{2} \)
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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

−16.18558451892084511429768125968, −15.14541879231397758206991882389, −13.82837766043971780716249445031, −12.44798190825503902997425963380, −11.01000508897486414710089415627, −8.927888807525103548494476663136, −6.89318754462180670469673449134, −5.85316934331565264126620218834, −2.81992434669930754527244503207, −0.24644143997199041334399091954, 3.50196931198950899694999990779, 4.46389247917359218091523782744, 7.37602071706597657884295976331, 9.173714265280191819795515625105, 10.73304257791744810850289928291, 12.29986408321535782742557226706, 13.43845668516149078223897932554, 15.39648911489713422707395786488, 16.52861213375434323453868277389, 17.19095454427168824350333113511

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