↑

Properties

Label 2-2e5-32.5-c3-0-7
Degree $2$
Conductor $32$
Sign $0.881 + 0.471i$
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  + (2.70 + 0.821i)2-s + (−3.28 − 7.93i)3-s + (6.65 + 4.44i)4-s + (11.2 + 4.67i)5-s + (−2.38 − 24.1i)6-s + (−11.8 − 11.8i)7-s + (14.3 + 17.4i)8-s + (−33.1 + 33.1i)9-s + (26.7 + 21.9i)10-s + (−23.5 + 56.9i)11-s + (13.4 − 67.4i)12-s + (−13.6 + 5.65i)13-s + (−22.3 − 41.8i)14-s − 104. i·15-s + (24.4 + 59.1i)16-s − 44.1i·17-s + ⋯
L(s)  = 1  + (0.956 + 0.290i)2-s + (−0.632 − 1.52i)3-s + (0.831 + 0.555i)4-s + (1.00 + 0.418i)5-s + (−0.161 − 1.64i)6-s + (−0.639 − 0.639i)7-s + (0.634 + 0.773i)8-s + (−1.22 + 1.22i)9-s + (0.844 + 0.693i)10-s + (−0.646 + 1.56i)11-s + (0.322 − 1.62i)12-s + (−0.291 + 0.120i)13-s + (−0.426 − 0.797i)14-s − 1.80i·15-s + (0.382 + 0.923i)16-s − 0.630i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.881 + 0.471i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.73863 - 0.435847i\)
\(L(\frac12)\) \(\approx\) \(1.73863 - 0.435847i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.70 - 0.821i)T \)
good3 \( 1 + (3.28 + 7.93i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-11.2 - 4.67i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (11.8 + 11.8i)T + 343iT^{2} \)
11 \( 1 + (23.5 - 56.9i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (13.6 - 5.65i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 44.1iT - 4.91e3T^{2} \)
19 \( 1 + (-66.5 + 27.5i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-60.0 + 60.0i)T - 1.21e4iT^{2} \)
29 \( 1 + (14.3 + 34.6i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 174.T + 2.97e4T^{2} \)
37 \( 1 + (118. + 49.0i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-15.5 + 15.5i)T - 6.89e4iT^{2} \)
43 \( 1 + (87.3 - 210. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 228. iT - 1.03e5T^{2} \)
53 \( 1 + (-258. + 624. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-456. - 188. i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-242. - 584. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-332. - 802. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (550. + 550. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-69.2 + 69.2i)T - 3.89e5iT^{2} \)
79 \( 1 + 518. iT - 4.93e5T^{2} \)
83 \( 1 + (595. - 246. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-656. - 656. i)T + 7.04e5iT^{2} \)
97 \( 1 + 388.T + 9.12e5T^{2} \)
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

−16.36947981218492903956361785199, −14.63075062387889977457468980892, −13.41805404661949881431123171790, −12.94091523844984834005208822668, −11.75966234644128660187955536710, −10.16610848612087589748042769060, −7.30449310924663460446359136360, −6.80346891862548116183938101178, −5.33529881466778885963094262971, −2.27355049005667563591970601265, 3.34312643032384377821948956663, 5.34953145961011669548263218932, 5.84489382417675392298705658497, 9.225652419933184343733194798732, 10.27729324951252588089834494825, 11.31097066698427435997570105432, 12.78711187017282638884231927411, 13.99268114167568778512123835390, 15.40185603872422910765841044636, 16.16375325799046794608177221936

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