Properties

Label 2-15-3.2-c10-0-7
Degree $2$
Conductor $15$
Sign $0.968 - 0.249i$
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  + 57.7i·2-s + (−60.6 − 235. i)3-s − 2.31e3·4-s − 1.39e3i·5-s + (1.35e4 − 3.50e3i)6-s + 2.27e4·7-s − 7.44e4i·8-s + (−5.17e4 + 2.85e4i)9-s + 8.07e4·10-s − 1.63e5i·11-s + (1.40e5 + 5.44e5i)12-s + 3.76e5·13-s + 1.31e6i·14-s + (−3.28e5 + 8.47e4i)15-s + 1.93e6·16-s − 1.43e6i·17-s + ⋯
L(s)  = 1  + 1.80i·2-s + (−0.249 − 0.968i)3-s − 2.25·4-s − 0.447i·5-s + (1.74 − 0.450i)6-s + 1.35·7-s − 2.27i·8-s + (−0.875 + 0.483i)9-s + 0.807·10-s − 1.01i·11-s + (0.563 + 2.18i)12-s + 1.01·13-s + 2.44i·14-s + (−0.433 + 0.111i)15-s + 1.84·16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.968 - 0.249i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.968 - 0.249i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.40332 + 0.177839i\)
\(L(\frac12)\) \(\approx\) \(1.40332 + 0.177839i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (60.6 + 235. i)T \)
5 \( 1 + 1.39e3iT \)
good2 \( 1 - 57.7iT - 1.02e3T^{2} \)
7 \( 1 - 2.27e4T + 2.82e8T^{2} \)
11 \( 1 + 1.63e5iT - 2.59e10T^{2} \)
13 \( 1 - 3.76e5T + 1.37e11T^{2} \)
17 \( 1 + 1.43e6iT - 2.01e12T^{2} \)
19 \( 1 - 1.20e6T + 6.13e12T^{2} \)
23 \( 1 + 4.68e6iT - 4.14e13T^{2} \)
29 \( 1 + 3.09e7iT - 4.20e14T^{2} \)
31 \( 1 - 3.58e7T + 8.19e14T^{2} \)
37 \( 1 + 8.77e7T + 4.80e15T^{2} \)
41 \( 1 - 6.35e7iT - 1.34e16T^{2} \)
43 \( 1 + 1.28e8T + 2.16e16T^{2} \)
47 \( 1 - 2.23e8iT - 5.25e16T^{2} \)
53 \( 1 + 8.77e6iT - 1.74e17T^{2} \)
59 \( 1 + 3.20e8iT - 5.11e17T^{2} \)
61 \( 1 + 3.58e8T + 7.13e17T^{2} \)
67 \( 1 + 2.46e8T + 1.82e18T^{2} \)
71 \( 1 - 1.70e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.99e9T + 4.29e18T^{2} \)
79 \( 1 - 1.11e9T + 9.46e18T^{2} \)
83 \( 1 + 3.65e9iT - 1.55e19T^{2} \)
89 \( 1 - 5.52e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.19e10T + 7.37e19T^{2} \)
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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

−16.95300387130890442970945613827, −15.79551751582540956909761557259, −14.15017915259298753944121897691, −13.52020345776346608125263002138, −11.54723448264019341496509072403, −8.606772110573595816779094837997, −7.86463598363825312651447981027, −6.25020489935746367749480104217, −4.98209737472531000494874735856, −0.804640946164658657927363600191, 1.61750743225960320590725281314, 3.59714470992768300190072446830, 4.95518056324085236225408406614, 8.645010569698581842836345798738, 10.19661635345746275853252975534, 11.03507866878977749149125683040, 12.05914454531596705497417201390, 13.92362929326670089352431559278, 15.17702888015268934844725608796, 17.45482420601081829460465220736

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