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Properties

Label 2-15-5.3-c10-0-3
Degree $2$
Conductor $15$
Sign $0.270 - 0.962i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (20.8 − 20.8i)2-s + (−99.2 − 99.2i)3-s + 157. i·4-s + (−1.51e3 + 2.73e3i)5-s − 4.12e3·6-s + (−6.93e3 + 6.93e3i)7-s + (2.45e4 + 2.45e4i)8-s + 1.96e4i·9-s + (2.53e4 + 8.84e4i)10-s − 1.59e4·11-s + (1.56e4 − 1.56e4i)12-s + (8.89e4 + 8.89e4i)13-s + 2.88e5i·14-s + (4.21e5 − 1.20e5i)15-s + 8.62e5·16-s + (−1.45e6 + 1.45e6i)17-s + ⋯
L(s)  = 1  + (0.650 − 0.650i)2-s + (−0.408 − 0.408i)3-s + 0.153i·4-s + (−0.484 + 0.874i)5-s − 0.531·6-s + (−0.412 + 0.412i)7-s + (0.750 + 0.750i)8-s + 0.333i·9-s + (0.253 + 0.884i)10-s − 0.0989·11-s + (0.0628 − 0.0628i)12-s + (0.239 + 0.239i)13-s + 0.536i·14-s + (0.554 − 0.159i)15-s + 0.822·16-s + (−1.02 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.270 - 0.962i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.270 - 0.962i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.10553 + 0.837369i\)
\(L(\frac12)\) \(\approx\) \(1.10553 + 0.837369i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (99.2 + 99.2i)T \)
5 \( 1 + (1.51e3 - 2.73e3i)T \)
good2 \( 1 + (-20.8 + 20.8i)T - 1.02e3iT^{2} \)
7 \( 1 + (6.93e3 - 6.93e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 1.59e4T + 2.59e10T^{2} \)
13 \( 1 + (-8.89e4 - 8.89e4i)T + 1.37e11iT^{2} \)
17 \( 1 + (1.45e6 - 1.45e6i)T - 2.01e12iT^{2} \)
19 \( 1 - 2.24e6iT - 6.13e12T^{2} \)
23 \( 1 + (3.90e6 + 3.90e6i)T + 4.14e13iT^{2} \)
29 \( 1 + 2.35e7iT - 4.20e14T^{2} \)
31 \( 1 - 9.89e6T + 8.19e14T^{2} \)
37 \( 1 + (8.26e7 - 8.26e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 2.02e8T + 1.34e16T^{2} \)
43 \( 1 + (-1.19e8 - 1.19e8i)T + 2.16e16iT^{2} \)
47 \( 1 + (-2.17e8 + 2.17e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (2.69e8 + 2.69e8i)T + 1.74e17iT^{2} \)
59 \( 1 - 4.93e8iT - 5.11e17T^{2} \)
61 \( 1 - 5.33e8T + 7.13e17T^{2} \)
67 \( 1 + (-1.67e9 + 1.67e9i)T - 1.82e18iT^{2} \)
71 \( 1 - 1.14e9T + 3.25e18T^{2} \)
73 \( 1 + (7.04e7 + 7.04e7i)T + 4.29e18iT^{2} \)
79 \( 1 - 3.90e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.92e9 + 1.92e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 5.81e9iT - 3.11e19T^{2} \)
97 \( 1 + (1.84e9 - 1.84e9i)T - 7.37e19iT^{2} \)
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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

−17.29819718737051848376332320571, −15.72869774312885433817465233029, −14.14026627914893117901803548536, −12.77451313096099673567509699933, −11.73497120901558211289318383308, −10.58348515138935978272052367944, −8.051227766209003371806065413188, −6.29816254150361750569724042512, −4.00848945693644428128302362871, −2.33841253185908367603976976509, 0.56700903944020665351827589705, 4.15687702753106623250662460214, 5.39987420992903987138378732681, 7.11535207719517813108066858751, 9.273918340816032750588666996571, 10.96100241477006907316886734343, 12.71358373143821320941731053148, 13.92712834998933032606846480123, 15.71850781731964223678714468508, 16.02290751506531941113047389489

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