↑

Properties

Label 2-2e5-32.13-c1-0-2
Degree $2$
Conductor $32$
Sign $0.576 + 0.817i$
Analytic cond. $0.255521$
Root an. cond. $0.505491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.443 − 1.34i)2-s + (0.0794 − 0.191i)3-s + (−1.60 + 1.19i)4-s + (0.707 − 0.292i)5-s + (−0.292 − 0.0215i)6-s + (−2.27 + 2.27i)7-s + (2.31 + 1.62i)8-s + (2.09 + 2.09i)9-s + (−0.707 − 0.819i)10-s + (−1.49 − 3.60i)11-s + (0.101 + 0.402i)12-s + (−4.50 − 1.86i)13-s + (4.05 + 2.04i)14-s − 0.158i·15-s + (1.15 − 3.82i)16-s − 3.05i·17-s + ⋯
L(s)  = 1  + (−0.313 − 0.949i)2-s + (0.0458 − 0.110i)3-s + (−0.803 + 0.595i)4-s + (0.316 − 0.130i)5-s + (−0.119 − 0.00880i)6-s + (−0.858 + 0.858i)7-s + (0.817 + 0.575i)8-s + (0.696 + 0.696i)9-s + (−0.223 − 0.259i)10-s + (−0.450 − 1.08i)11-s + (0.0291 + 0.116i)12-s + (−1.24 − 0.517i)13-s + (1.08 + 0.545i)14-s − 0.0410i·15-s + (0.289 − 0.957i)16-s − 0.740i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.576 + 0.817i$
Analytic conductor: \(0.255521\)
Root analytic conductor: \(0.505491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1/2),\ 0.576 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534429 - 0.277037i\)
\(L(\frac12)\) \(\approx\) \(0.534429 - 0.277037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.443 + 1.34i)T \)
good3 \( 1 + (-0.0794 + 0.191i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.707 + 0.292i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.27 - 2.27i)T - 7iT^{2} \)
11 \( 1 + (1.49 + 3.60i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (4.50 + 1.86i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 3.05iT - 17T^{2} \)
19 \( 1 + (-3.87 - 1.60i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.271 - 0.271i)T + 23iT^{2} \)
29 \( 1 + (0.931 - 2.24i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (-3.63 + 1.50i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.54 + 1.54i)T + 41iT^{2} \)
43 \( 1 + (-0.748 - 1.80i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.37iT - 47T^{2} \)
53 \( 1 + (-1.67 - 4.04i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (10.1 - 4.19i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.35 + 3.28i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.99 - 4.81i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-6.47 + 6.47i)T - 71iT^{2} \)
73 \( 1 + (2.84 + 2.84i)T + 73iT^{2} \)
79 \( 1 - 9.74iT - 79T^{2} \)
83 \( 1 + (9.04 + 3.74i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-7.58 + 7.58i)T - 89iT^{2} \)
97 \( 1 - 3.71T + 97T^{2} \)
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

−16.82682638086308839166579305643, −15.74643805891306379893689329844, −13.81950652636914328114019056467, −12.91647944403095541967177358666, −11.82580475248531804649755308070, −10.25908696994540061538134632247, −9.305020759521921105361772097402, −7.72735097554307607334510753929, −5.30786678338673466944267923995, −2.82074860538497551180743379716, 4.45272389521002682795052947765, 6.53469146775854751516330082994, 7.51718548353460830738786687037, 9.674335322942448867537047349946, 10.02487162970451494527276710368, 12.50852223399705057322881970747, 13.66446509829547983844827284940, 14.92185428459844273959238902396, 15.87006652051185973190131434464, 17.09366087832283902287578884224

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