Properties

Label 2-15-5.3-c10-0-1
Degree $2$
Conductor $15$
Sign $-0.134 + 0.990i$
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  + (−32.7 + 32.7i)2-s + (99.2 + 99.2i)3-s − 1.12e3i·4-s + (−1.11e3 + 2.91e3i)5-s − 6.50e3·6-s + (−1.60e4 + 1.60e4i)7-s + (3.39e3 + 3.39e3i)8-s + 1.96e4i·9-s + (−5.89e4 − 1.32e5i)10-s + 2.46e5·11-s + (1.11e5 − 1.11e5i)12-s + (−4.63e5 − 4.63e5i)13-s − 1.05e6i·14-s + (−4.00e5 + 1.78e5i)15-s + 9.31e5·16-s + (2.21e5 − 2.21e5i)17-s + ⋯
L(s)  = 1  + (−1.02 + 1.02i)2-s + (0.408 + 0.408i)3-s − 1.10i·4-s + (−0.358 + 0.933i)5-s − 0.836·6-s + (−0.955 + 0.955i)7-s + (0.103 + 0.103i)8-s + 0.333i·9-s + (−0.589 − 1.32i)10-s + 1.53·11-s + (0.449 − 0.449i)12-s + (−1.24 − 1.24i)13-s − 1.95i·14-s + (−0.527 + 0.234i)15-s + 0.888·16-s + (0.156 − 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.134 + 0.990i$
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: \((2,\ 15,\ (\ :5),\ -0.134 + 0.990i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.231936 - 0.265425i\)
\(L(\frac12)\) \(\approx\) \(0.231936 - 0.265425i\)
\(L(6)\) 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 \( 1 + (1.11e3 - 2.91e3i)T \)
good2 \( 1 + (32.7 - 32.7i)T - 1.02e3iT^{2} \)
7 \( 1 + (1.60e4 - 1.60e4i)T - 2.82e8iT^{2} \)
11 \( 1 - 2.46e5T + 2.59e10T^{2} \)
13 \( 1 + (4.63e5 + 4.63e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-2.21e5 + 2.21e5i)T - 2.01e12iT^{2} \)
19 \( 1 - 5.94e5iT - 6.13e12T^{2} \)
23 \( 1 + (6.64e5 + 6.64e5i)T + 4.14e13iT^{2} \)
29 \( 1 + 3.97e6iT - 4.20e14T^{2} \)
31 \( 1 + 2.75e7T + 8.19e14T^{2} \)
37 \( 1 + (4.26e7 - 4.26e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.16e7T + 1.34e16T^{2} \)
43 \( 1 + (-7.66e7 - 7.66e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.67e8 - 1.67e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-4.00e7 - 4.00e7i)T + 1.74e17iT^{2} \)
59 \( 1 - 1.24e9iT - 5.11e17T^{2} \)
61 \( 1 + 7.61e8T + 7.13e17T^{2} \)
67 \( 1 + (-1.02e9 + 1.02e9i)T - 1.82e18iT^{2} \)
71 \( 1 + 2.68e9T + 3.25e18T^{2} \)
73 \( 1 + (2.06e9 + 2.06e9i)T + 4.29e18iT^{2} \)
79 \( 1 + 3.33e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.16e9 - 2.16e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 1.22e9iT - 3.11e19T^{2} \)
97 \( 1 + (9.60e9 - 9.60e9i)T - 7.37e19iT^{2} \)
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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

−17.76382849234478423591365495439, −16.46036495071566826354624225832, −15.27295388559409351251139570394, −14.64616752422983512949870056908, −12.21904472899920235334745328091, −10.07731681164548941169597735351, −9.081083759096832891962951994274, −7.50941006033491810697787385406, −6.14598593805598253156294383120, −3.14596976807950087900078175282, 0.22660215466526854692490098661, 1.62149924101903553313415659097, 3.81673693873170056208993612852, 7.11218491365167385492272196643, 8.938024678996580922980159259134, 9.729760546737921552096284950209, 11.65248977975128628995766335843, 12.64474830092841739237580873628, 14.30083480592769759543590458808, 16.58988111030326479569807814158

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