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Properties

Label 2-2e5-32.21-c3-0-10
Degree $2$
Conductor $32$
Sign $0.395 + 0.918i$
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  + (1.17 − 2.57i)2-s + (7.57 − 3.13i)3-s + (−5.22 − 6.05i)4-s + (−8.03 + 19.4i)5-s + (0.844 − 23.1i)6-s + (−1.85 − 1.85i)7-s + (−21.7 + 6.32i)8-s + (28.3 − 28.3i)9-s + (40.4 + 43.5i)10-s + (8.63 + 3.57i)11-s + (−58.5 − 29.4i)12-s + (−11.5 − 27.9i)13-s + (−6.93 + 2.58i)14-s + 172. i·15-s + (−9.29 + 63.3i)16-s + 7.99i·17-s + ⋯
L(s)  = 1  + (0.416 − 0.909i)2-s + (1.45 − 0.603i)3-s + (−0.653 − 0.756i)4-s + (−0.718 + 1.73i)5-s + (0.0574 − 1.57i)6-s + (−0.0999 − 0.0999i)7-s + (−0.960 + 0.279i)8-s + (1.05 − 1.05i)9-s + (1.27 + 1.37i)10-s + (0.236 + 0.0979i)11-s + (−1.40 − 0.707i)12-s + (−0.247 − 0.597i)13-s + (−0.132 + 0.0492i)14-s + 2.96i·15-s + (−0.145 + 0.989i)16-s + 0.114i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.395 + 0.918i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.52028 - 1.00043i\)
\(L(\frac12)\) \(\approx\) \(1.52028 - 1.00043i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 2.57i)T \)
good3 \( 1 + (-7.57 + 3.13i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (8.03 - 19.4i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (1.85 + 1.85i)T + 343iT^{2} \)
11 \( 1 + (-8.63 - 3.57i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (11.5 + 27.9i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 7.99iT - 4.91e3T^{2} \)
19 \( 1 + (-5.76 - 13.9i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-60.2 + 60.2i)T - 1.21e4iT^{2} \)
29 \( 1 + (-167. + 69.3i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 225.T + 2.97e4T^{2} \)
37 \( 1 + (-0.431 + 1.04i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (275. - 275. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-257. - 106. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 51.3iT - 1.03e5T^{2} \)
53 \( 1 + (2.98 + 1.23i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-101. + 244. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (270. - 111. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-778. + 322. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-484. - 484. i)T + 3.57e5iT^{2} \)
73 \( 1 + (212. - 212. i)T - 3.89e5iT^{2} \)
79 \( 1 - 593. iT - 4.93e5T^{2} \)
83 \( 1 + (-320. - 773. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-435. - 435. i)T + 7.04e5iT^{2} \)
97 \( 1 + 570.T + 9.12e5T^{2} \)
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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

−15.25371219377940858655527268169, −14.64018123307421723179504876479, −13.83257193459346695911432441726, −12.53897219943770264230987195492, −11.11340150242056307482098554184, −9.889528498121141447559410477642, −8.191980276840900434330786687729, −6.81748616691275363812883213561, −3.61934487115080960383027731883, −2.58661485736058550420121867050, 3.77457883727207504260897840162, 4.94665738401804836568429794347, 7.59504536161378391696920565693, 8.783915345291148181353536776024, 9.237682105252411363914649952159, 12.12462321698778305725213156563, 13.23337121484621211728903596418, 14.28515467465189308963249721335, 15.49200328614128213330487761387, 16.11794100285676904627588744963

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