↑

Properties

Label 2-6-3.2-c30-0-0
Degree $2$
Conductor $6$
Sign $-0.849 - 0.528i$
Analytic cond. $34.2085$
Root an. cond. $5.84880$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.31e4i·2-s + (7.57e6 − 1.21e7i)3-s − 5.36e8·4-s − 2.86e10i·5-s + (2.82e11 + 1.75e11i)6-s − 3.86e12·7-s − 1.24e13i·8-s + (−9.10e13 − 1.84e14i)9-s + 6.64e14·10-s − 1.21e15i·11-s + (−4.06e15 + 6.54e15i)12-s − 5.72e16·13-s − 8.96e16i·14-s + (−3.49e17 − 2.17e17i)15-s + 2.88e17·16-s + 4.94e18i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.528 − 0.849i)3-s − 0.500·4-s − 0.939i·5-s + (0.600 + 0.373i)6-s − 0.815·7-s − 0.353i·8-s + (−0.442 − 0.896i)9-s + 0.664·10-s − 0.291i·11-s + (−0.264 + 0.424i)12-s − 1.11·13-s − 0.576i·14-s + (−0.797 − 0.496i)15-s + 0.250·16-s + 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.849 - 0.528i$
Analytic conductor: \(34.2085\)
Root analytic conductor: \(5.84880\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :15),\ -0.849 - 0.528i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(0.1301056120\)
\(L(\frac12)\) \(\approx\) \(0.1301056120\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.31e4iT \)
3 \( 1 + (-7.57e6 + 1.21e7i)T \)
good5 \( 1 + 2.86e10iT - 9.31e20T^{2} \)
7 \( 1 + 3.86e12T + 2.25e25T^{2} \)
11 \( 1 + 1.21e15iT - 1.74e31T^{2} \)
13 \( 1 + 5.72e16T + 2.61e33T^{2} \)
17 \( 1 - 4.94e18iT - 8.19e36T^{2} \)
19 \( 1 - 2.15e19T + 2.30e38T^{2} \)
23 \( 1 - 1.53e20iT - 7.10e40T^{2} \)
29 \( 1 - 8.01e21iT - 7.44e43T^{2} \)
31 \( 1 + 2.10e22T + 5.50e44T^{2} \)
37 \( 1 + 1.99e23T + 1.11e47T^{2} \)
41 \( 1 + 2.99e23iT - 2.41e48T^{2} \)
43 \( 1 + 5.08e24T + 1.00e49T^{2} \)
47 \( 1 - 5.51e24iT - 1.45e50T^{2} \)
53 \( 1 + 2.02e25iT - 5.34e51T^{2} \)
59 \( 1 + 4.17e26iT - 1.33e53T^{2} \)
61 \( 1 - 1.01e27T + 3.62e53T^{2} \)
67 \( 1 + 4.41e27T + 6.05e54T^{2} \)
71 \( 1 - 8.92e27iT - 3.44e55T^{2} \)
73 \( 1 + 4.52e27T + 7.93e55T^{2} \)
79 \( 1 - 3.16e28T + 8.48e56T^{2} \)
83 \( 1 + 2.37e27iT - 3.73e57T^{2} \)
89 \( 1 - 5.31e28iT - 3.03e58T^{2} \)
97 \( 1 + 6.93e29T + 4.01e59T^{2} \)
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

−16.32004503925332440816084911806, −14.71076588779412808269138704519, −13.23228082187117726619226987457, −12.35095680566391364553051362139, −9.537497126788520310399275532938, −8.318518094844118365212004320032, −6.95718889888448078207686344118, −5.43853366616686656211622215531, −3.42236521083788146418860987604, −1.41386675328181599449966430386, 0.03624146001042523373188217199, 2.52656601871406933873072569557, 3.29597723625859225392928034442, 4.98497270617219219548418578398, 7.27249566444629103518211034142, 9.399716590603060903892387599683, 10.18557855213777431421793049163, 11.73360348940803673988956889490, 13.67492915765602090285829373202, 14.85294044092263865168257354813

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