Properties

Label 2-2e5-32.21-c3-0-1
Degree $2$
Conductor $32$
Sign $-0.953 - 0.300i$
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  + (−0.719 + 2.73i)2-s + (−2.66 + 1.10i)3-s + (−6.96 − 3.93i)4-s + (−5.50 + 13.2i)5-s + (−1.09 − 8.07i)6-s + (−6.48 − 6.48i)7-s + (15.7 − 16.2i)8-s + (−13.2 + 13.2i)9-s + (−32.4 − 24.6i)10-s + (49.3 + 20.4i)11-s + (22.8 + 2.80i)12-s + (21.4 + 51.7i)13-s + (22.3 − 13.0i)14-s − 41.4i·15-s + (32.9 + 54.8i)16-s + 3.73i·17-s + ⋯
L(s)  = 1  + (−0.254 + 0.967i)2-s + (−0.512 + 0.212i)3-s + (−0.870 − 0.492i)4-s + (−0.492 + 1.18i)5-s + (−0.0748 − 0.549i)6-s + (−0.349 − 0.349i)7-s + (0.697 − 0.716i)8-s + (−0.489 + 0.489i)9-s + (−1.02 − 0.779i)10-s + (1.35 + 0.560i)11-s + (0.550 + 0.0675i)12-s + (0.457 + 1.10i)13-s + (0.427 − 0.249i)14-s − 0.714i·15-s + (0.515 + 0.857i)16-s + 0.0533i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.953 - 0.300i$
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: \((2,\ 32,\ (\ :3/2),\ -0.953 - 0.300i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.101406 + 0.659553i\)
\(L(\frac12)\) \(\approx\) \(0.101406 + 0.659553i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.719 - 2.73i)T \)
good3 \( 1 + (2.66 - 1.10i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (5.50 - 13.2i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (6.48 + 6.48i)T + 343iT^{2} \)
11 \( 1 + (-49.3 - 20.4i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (-21.4 - 51.7i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 3.73iT - 4.91e3T^{2} \)
19 \( 1 + (36.7 + 88.6i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (45.4 - 45.4i)T - 1.21e4iT^{2} \)
29 \( 1 + (-51.9 + 21.5i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 73.5T + 2.97e4T^{2} \)
37 \( 1 + (165. - 399. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-334. + 334. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-328. - 136. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 185. iT - 1.03e5T^{2} \)
53 \( 1 + (-412. - 171. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (214. - 518. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-85.1 + 35.2i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (252. - 104. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-430. - 430. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-41.8 + 41.8i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.21e3iT - 4.93e5T^{2} \)
83 \( 1 + (290. + 702. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-365. - 365. i)T + 7.04e5iT^{2} \)
97 \( 1 - 508.T + 9.12e5T^{2} \)
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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

−16.84960908290817804711196106758, −15.75896218896751115663074647233, −14.61448061902420904843230413714, −13.74875248281394277942116699101, −11.65573535442990046649162519253, −10.52875270156405510411248039154, −9.040404772513349018412857247018, −7.20696092992906373472844546353, −6.32558210071226135514242247513, −4.20903388743908697347610181018, 0.75557527514635534891477809551, 3.80797267149740981085072869820, 5.77820694416192809280955622549, 8.332629157434290016514393379108, 9.224275356798787394487580415246, 10.99337882560958276445894828602, 12.22842333549906655386245685935, 12.62722067118835635107276061533, 14.33076272507482594823640461630, 16.22712710172602104213843275789

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