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Properties

Label 2-2e5-32.5-c3-0-3
Degree $2$
Conductor $32$
Sign $0.979 - 0.203i$
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  + (−2.51 + 1.28i)2-s + (−0.998 − 2.40i)3-s + (4.68 − 6.48i)4-s + (17.4 + 7.20i)5-s + (5.61 + 4.78i)6-s + (4.37 + 4.37i)7-s + (−3.42 + 22.3i)8-s + (14.2 − 14.2i)9-s + (−53.0 + 4.26i)10-s + (11.7 − 28.4i)11-s + (−20.3 − 4.80i)12-s + (−12.9 + 5.35i)13-s + (−16.6 − 5.37i)14-s − 49.1i·15-s + (−20.1 − 60.7i)16-s + 72.9i·17-s + ⋯
L(s)  = 1  + (−0.890 + 0.455i)2-s + (−0.192 − 0.463i)3-s + (0.585 − 0.810i)4-s + (1.55 + 0.644i)5-s + (0.382 + 0.325i)6-s + (0.236 + 0.236i)7-s + (−0.151 + 0.988i)8-s + (0.528 − 0.528i)9-s + (−1.67 + 0.134i)10-s + (0.323 − 0.780i)11-s + (−0.488 − 0.115i)12-s + (−0.275 + 0.114i)13-s + (−0.317 − 0.102i)14-s − 0.845i·15-s + (−0.315 − 0.948i)16-s + 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(1.88806\)
Root analytic conductor: \(1.37406\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :3/2),\ 0.979 - 0.203i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.993084 + 0.101952i\)
\(L(\frac12)\) \(\approx\) \(0.993084 + 0.101952i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.51 - 1.28i)T \)
good3 \( 1 + (0.998 + 2.40i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-17.4 - 7.20i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (-4.37 - 4.37i)T + 343iT^{2} \)
11 \( 1 + (-11.7 + 28.4i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (12.9 - 5.35i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 72.9iT - 4.91e3T^{2} \)
19 \( 1 + (143. - 59.2i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-83.6 + 83.6i)T - 1.21e4iT^{2} \)
29 \( 1 + (39.6 + 95.7i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 29.0T + 2.97e4T^{2} \)
37 \( 1 + (267. + 110. i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (124. - 124. i)T - 6.89e4iT^{2} \)
43 \( 1 + (27.0 - 65.2i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 282. iT - 1.03e5T^{2} \)
53 \( 1 + (51.4 - 124. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (222. + 92.1i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (226. + 547. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-356. - 859. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-690. - 690. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-223. + 223. i)T - 3.89e5iT^{2} \)
79 \( 1 + 698. iT - 4.93e5T^{2} \)
83 \( 1 + (915. - 379. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (163. + 163. i)T + 7.04e5iT^{2} \)
97 \( 1 + 839.T + 9.12e5T^{2} \)
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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

−16.92545131344268339323582970289, −15.08456494792990519956845927376, −14.24273998845943721572696973031, −12.75956648731267748380477386011, −10.92698313480240511944162649582, −9.919695557075536631126047926701, −8.600600405188530970458773815662, −6.68352411913329630701818097405, −5.96327748169301195515191182444, −1.85877497140206263464220862348, 1.88822100586251042606672902555, 4.89108578321998216471637731789, 6.99873774849215537388518198377, 8.961353734946419372521456117937, 9.854273384997515206968051761345, 10.82759577892749891564936794540, 12.56037025157248069557287782514, 13.57350116947089950257076170717, 15.40676194721134848374296355932, 16.90044855690779481615998812864

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