Properties

Label 2-2e5-32.13-c3-0-2
Degree $2$
Conductor $32$
Sign $-0.901 - 0.433i$
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  + (−0.582 + 2.76i)2-s + (−1.90 + 4.59i)3-s + (−7.32 − 3.22i)4-s + (0.188 − 0.0782i)5-s + (−11.5 − 7.93i)6-s + (−11.4 + 11.4i)7-s + (13.1 − 18.3i)8-s + (1.63 + 1.63i)9-s + (0.106 + 0.568i)10-s + (18.6 + 45.0i)11-s + (28.7 − 27.4i)12-s + (18.9 + 7.84i)13-s + (−24.9 − 38.2i)14-s + 1.01i·15-s + (43.2 + 47.2i)16-s − 85.7i·17-s + ⋯
L(s)  = 1  + (−0.205 + 0.978i)2-s + (−0.365 + 0.883i)3-s + (−0.915 − 0.403i)4-s + (0.0168 − 0.00699i)5-s + (−0.789 − 0.540i)6-s + (−0.616 + 0.616i)7-s + (0.582 − 0.812i)8-s + (0.0603 + 0.0603i)9-s + (0.00336 + 0.0179i)10-s + (0.511 + 1.23i)11-s + (0.691 − 0.661i)12-s + (0.404 + 0.167i)13-s + (−0.476 − 0.729i)14-s + 0.0174i·15-s + (0.675 + 0.737i)16-s − 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.901 - 0.433i$
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: \((2,\ 32,\ (\ :3/2),\ -0.901 - 0.433i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.189687 + 0.832571i\)
\(L(\frac12)\) \(\approx\) \(0.189687 + 0.832571i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.582 - 2.76i)T \)
good3 \( 1 + (1.90 - 4.59i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-0.188 + 0.0782i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (11.4 - 11.4i)T - 343iT^{2} \)
11 \( 1 + (-18.6 - 45.0i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (-18.9 - 7.84i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 85.7iT - 4.91e3T^{2} \)
19 \( 1 + (-110. - 45.8i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (74.2 + 74.2i)T + 1.21e4iT^{2} \)
29 \( 1 + (-64.4 + 155. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 36.6T + 2.97e4T^{2} \)
37 \( 1 + (-313. + 129. i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-196. - 196. i)T + 6.89e4iT^{2} \)
43 \( 1 + (20.8 + 50.4i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 508. iT - 1.03e5T^{2} \)
53 \( 1 + (-73.8 - 178. i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-40.9 + 16.9i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (324. - 784. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (49.4 - 119. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (362. - 362. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-239. - 239. i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.01e3iT - 4.93e5T^{2} \)
83 \( 1 + (-231. - 95.7i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-1.10e3 + 1.10e3i)T - 7.04e5iT^{2} \)
97 \( 1 + 74.0T + 9.12e5T^{2} \)
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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

−16.43324393405320066360622863983, −15.87376288717006679846883148961, −14.86998862036512733301586914264, −13.48501918546745398648581218712, −11.89022843119530898994584597735, −9.951533904626397806583832542506, −9.369824855308131694589413687916, −7.43703344025802092726433042687, −5.81431832370542175535275595627, −4.36581229847544149202743941020, 0.987826990991300641736974864116, 3.61635548842420196003260369288, 6.17897157018514410109151848691, 7.926748370695143541585642428463, 9.544221690461296875674237818706, 10.95640719241665891850732617905, 12.06621483341243460820950998720, 13.18362508303525506817486454573, 13.96187166919176123451146802439, 16.14974716682393866863525081824

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