↑

Properties

Label 2-2e5-32.29-c3-0-6
Degree $2$
Conductor $32$
Sign $0.992 - 0.119i$
Analytic cond. $1.88806$
Root an. cond. $1.37406$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 + 0.264i)2-s + (0.143 + 0.0595i)3-s + (7.85 + 1.49i)4-s + (−0.767 − 1.85i)5-s + (0.389 + 0.205i)6-s + (−5.47 + 5.47i)7-s + (21.7 + 6.28i)8-s + (−19.0 − 19.0i)9-s + (−1.67 − 5.42i)10-s + (−36.9 + 15.2i)11-s + (1.04 + 0.682i)12-s + (4.49 − 10.8i)13-s + (−16.8 + 13.9i)14-s − 0.312i·15-s + (59.5 + 23.4i)16-s − 53.8i·17-s + ⋯
L(s)  = 1  + (0.995 + 0.0936i)2-s + (0.0276 + 0.0114i)3-s + (0.982 + 0.186i)4-s + (−0.0686 − 0.165i)5-s + (0.0264 + 0.0140i)6-s + (−0.295 + 0.295i)7-s + (0.960 + 0.277i)8-s + (−0.706 − 0.706i)9-s + (−0.0528 − 0.171i)10-s + (−1.01 + 0.419i)11-s + (0.0250 + 0.0164i)12-s + (0.0959 − 0.231i)13-s + (−0.322 + 0.266i)14-s − 0.00537i·15-s + (0.930 + 0.366i)16-s − 0.768i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.992 - 0.119i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.992 - 0.119i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.94488 + 0.116770i\)
\(L(\frac12)\) \(\approx\) \(1.94488 + 0.116770i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 - 0.264i)T \)
good3 \( 1 + (-0.143 - 0.0595i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (0.767 + 1.85i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (5.47 - 5.47i)T - 343iT^{2} \)
11 \( 1 + (36.9 - 15.2i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-4.49 + 10.8i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 53.8iT - 4.91e3T^{2} \)
19 \( 1 + (31.9 - 77.2i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-50.0 - 50.0i)T + 1.21e4iT^{2} \)
29 \( 1 + (-156. - 64.6i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 207.T + 2.97e4T^{2} \)
37 \( 1 + (-73.7 - 177. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (293. + 293. i)T + 6.89e4iT^{2} \)
43 \( 1 + (342. - 141. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 510. iT - 1.03e5T^{2} \)
53 \( 1 + (590. - 244. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (257. + 622. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-424. - 175. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-482. - 199. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (199. - 199. i)T - 3.57e5iT^{2} \)
73 \( 1 + (127. + 127. i)T + 3.89e5iT^{2} \)
79 \( 1 + 237. iT - 4.93e5T^{2} \)
83 \( 1 + (-9.62 + 23.2i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-329. + 329. i)T - 7.04e5iT^{2} \)
97 \( 1 - 776.T + 9.12e5T^{2} \)
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

−15.99398305958114319928013734555, −15.12039421306160725082682403809, −13.93319213399176048433041546933, −12.67831784177951025895548935635, −11.76933858947901494207391279317, −10.21249814358124544316741883981, −8.267016427627586221233845844485, −6.53560433994269969379202019481, −5.05895247302539925409716821995, −3.00747629514602574133697408801, 2.86972154817773962168170599075, 4.91687636348170795679488693282, 6.51130733864453223682030991596, 8.167043463396753798926763624463, 10.43883012008113331949308251085, 11.33159297739783578584330585777, 12.92882393282432613323382079925, 13.72409689916140394978741509904, 14.97141762824621390882051120550, 16.09187742591124759730846088525

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