↑

Properties

Label 2-15-15.14-c10-0-10
Degree $2$
Conductor $15$
Sign $0.620 + 0.784i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.37·2-s + (−224. + 92.2i)3-s − 1.01e3·4-s + (862. + 3.00e3i)5-s + (758. − 311. i)6-s − 1.60e4i·7-s + 6.87e3·8-s + (4.20e4 − 4.14e4i)9-s + (−2.90e3 − 1.01e4i)10-s + 1.09e5i·11-s + (2.27e5 − 9.33e4i)12-s − 5.72e5i·13-s + 5.40e4i·14-s + (−4.70e5 − 5.95e5i)15-s + 1.01e6·16-s + 1.54e6·17-s + ⋯
L(s)  = 1  − 0.105·2-s + (−0.925 + 0.379i)3-s − 0.988·4-s + (0.275 + 0.961i)5-s + (0.0975 − 0.0400i)6-s − 0.952i·7-s + 0.209·8-s + (0.711 − 0.702i)9-s + (−0.0290 − 0.101i)10-s + 0.677i·11-s + (0.914 − 0.375i)12-s − 1.54i·13-s + 0.100i·14-s + (−0.620 − 0.784i)15-s + 0.966·16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.620 + 0.784i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.620 + 0.784i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.679946 - 0.329247i\)
\(L(\frac12)\) \(\approx\) \(0.679946 - 0.329247i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (224. - 92.2i)T \)
5 \( 1 + (-862. - 3.00e3i)T \)
good2 \( 1 + 3.37T + 1.02e3T^{2} \)
7 \( 1 + 1.60e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.09e5iT - 2.59e10T^{2} \)
13 \( 1 + 5.72e5iT - 1.37e11T^{2} \)
17 \( 1 - 1.54e6T + 2.01e12T^{2} \)
19 \( 1 + 7.20e5T + 6.13e12T^{2} \)
23 \( 1 + 2.49e6T + 4.14e13T^{2} \)
29 \( 1 + 1.79e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.83e7T + 8.19e14T^{2} \)
37 \( 1 + 6.73e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.78e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.66e8iT - 2.16e16T^{2} \)
47 \( 1 - 2.99e8T + 5.25e16T^{2} \)
53 \( 1 + 6.37e8T + 1.74e17T^{2} \)
59 \( 1 - 3.05e7iT - 5.11e17T^{2} \)
61 \( 1 - 9.80e8T + 7.13e17T^{2} \)
67 \( 1 + 6.91e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.10e9iT - 3.25e18T^{2} \)
73 \( 1 + 2.47e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.45e9T + 9.46e18T^{2} \)
83 \( 1 + 6.85e9T + 1.55e19T^{2} \)
89 \( 1 - 5.45e9iT - 3.11e19T^{2} \)
97 \( 1 - 3.04e9iT - 7.37e19T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.20791966458878886821644212726, −15.33865228031359967496405596694, −13.98982512967373428221704120338, −12.52590957561080803878319495510, −10.55011236051144751347166772890, −9.950037919438759961383179266040, −7.49446728565877140730108569606, −5.60251009103338649906738941556, −3.86022106033983278651077296113, −0.51813608436055813145805537791, 1.25206584676147868705059700726, 4.68431075936546686306319376045, 5.94171716486483719831424804731, 8.383926861513862722085031107957, 9.663541693088509391844865651568, 11.75345476743349804713254804980, 12.78731806866124908896315333021, 14.06298746777010434317927593105, 16.25160109493079726306442712205, 17.07907105934383281385461694729

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