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Properties

Label 2-2e5-32.13-c3-0-10
Degree $2$
Conductor $32$
Sign $-0.693 + 0.720i$
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  + (−1.75 − 2.21i)2-s + (3.54 − 8.56i)3-s + (−1.80 + 7.79i)4-s + (−7.55 + 3.12i)5-s + (−25.2 + 7.21i)6-s + (7.16 − 7.16i)7-s + (20.4 − 9.71i)8-s + (−41.6 − 41.6i)9-s + (20.2 + 11.2i)10-s + (0.758 + 1.83i)11-s + (60.3 + 43.1i)12-s + (71.0 + 29.4i)13-s + (−28.4 − 3.25i)14-s + 75.7i·15-s + (−57.4 − 28.1i)16-s − 98.5i·17-s + ⋯
L(s)  = 1  + (−0.622 − 0.782i)2-s + (0.682 − 1.64i)3-s + (−0.225 + 0.974i)4-s + (−0.675 + 0.279i)5-s + (−1.71 + 0.491i)6-s + (0.386 − 0.386i)7-s + (0.903 − 0.429i)8-s + (−1.54 − 1.54i)9-s + (0.639 + 0.354i)10-s + (0.0207 + 0.0501i)11-s + (1.45 + 1.03i)12-s + (1.51 + 0.628i)13-s + (−0.543 − 0.0621i)14-s + 1.30i·15-s + (−0.897 − 0.440i)16-s − 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.693 + 0.720i$
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: \((2,\ 32,\ (\ :3/2),\ -0.693 + 0.720i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.398437 - 0.936032i\)
\(L(\frac12)\) \(\approx\) \(0.398437 - 0.936032i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.75 + 2.21i)T \)
good3 \( 1 + (-3.54 + 8.56i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (7.55 - 3.12i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-7.16 + 7.16i)T - 343iT^{2} \)
11 \( 1 + (-0.758 - 1.83i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (-71.0 - 29.4i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 98.5iT - 4.91e3T^{2} \)
19 \( 1 + (-89.5 - 37.1i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-24.9 - 24.9i)T + 1.21e4iT^{2} \)
29 \( 1 + (57.8 - 139. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 58.0T + 2.97e4T^{2} \)
37 \( 1 + (202. - 84.0i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (45.3 + 45.3i)T + 6.89e4iT^{2} \)
43 \( 1 + (-89.7 - 216. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 4.38iT - 1.03e5T^{2} \)
53 \( 1 + (-8.98 - 21.6i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-287. + 119. i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (28.2 - 68.0i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-293. + 708. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (579. - 579. i)T - 3.57e5iT^{2} \)
73 \( 1 + (258. + 258. i)T + 3.89e5iT^{2} \)
79 \( 1 + 834. iT - 4.93e5T^{2} \)
83 \( 1 + (-234. - 97.3i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-179. + 179. i)T - 7.04e5iT^{2} \)
97 \( 1 + 624.T + 9.12e5T^{2} \)
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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

−16.09085676260718737697195498829, −14.08119955305064280214541390470, −13.38996045398176414234564638262, −11.98963957270148087276211258049, −11.24613209294271769852071867247, −9.087486015236727422771758795527, −7.891193103938811729445767073998, −7.04203684674457329487074778518, −3.32811322040397881053412948614, −1.30372120499367736090782831080, 3.97163071544693128048303922007, 5.55100128875668613272421126621, 8.161906414596607016740470031595, 8.773616179310064238316057533863, 10.16657194814568425439210926410, 11.21739164003723434448970253987, 13.71100538056522450215504050956, 15.02078184408997633038651327843, 15.58422515473722607382706052489, 16.31488979135728816733953225036

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