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Properties

Label 2-2e5-32.13-c3-0-5
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

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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: $\chi_{32} (13, \cdot )$
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 \( 1 + 388.T + 9.12e5T^{2} \)
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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

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

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