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Properties

Label 2-2e8-32.21-c3-0-11
Degree $2$
Conductor $256$
Sign $0.829 - 0.558i$
Analytic cond. $15.1044$
Root an. cond. $3.88644$
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  + (4.56 − 1.89i)3-s + (−1.37 + 3.30i)5-s + (6.14 + 6.14i)7-s + (−1.79 + 1.79i)9-s + (17.2 + 7.14i)11-s + (25.9 + 62.7i)13-s + 17.7i·15-s − 87.5i·17-s + (48.8 + 117. i)19-s + (39.7 + 16.4i)21-s + (55.7 − 55.7i)23-s + (79.3 + 79.3i)25-s + (−55.9 + 134. i)27-s + (114. − 47.2i)29-s − 229.·31-s + ⋯
L(s)  = 1  + (0.879 − 0.364i)3-s + (−0.122 + 0.295i)5-s + (0.331 + 0.331i)7-s + (−0.0665 + 0.0665i)9-s + (0.472 + 0.195i)11-s + (0.554 + 1.33i)13-s + 0.304i·15-s − 1.24i·17-s + (0.589 + 1.42i)19-s + (0.412 + 0.170i)21-s + (0.505 − 0.505i)23-s + (0.634 + 0.634i)25-s + (−0.398 + 0.962i)27-s + (0.730 − 0.302i)29-s − 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ 0.829 - 0.558i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.470265384\)
\(L(\frac12)\) \(\approx\) \(2.470265384\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-4.56 + 1.89i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (1.37 - 3.30i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-6.14 - 6.14i)T + 343iT^{2} \)
11 \( 1 + (-17.2 - 7.14i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (-25.9 - 62.7i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 87.5iT - 4.91e3T^{2} \)
19 \( 1 + (-48.8 - 117. i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-55.7 + 55.7i)T - 1.21e4iT^{2} \)
29 \( 1 + (-114. + 47.2i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 229.T + 2.97e4T^{2} \)
37 \( 1 + (-123. + 298. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-111. + 111. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-76.5 - 31.7i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 367. iT - 1.03e5T^{2} \)
53 \( 1 + (-244. - 101. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (183. - 442. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (524. - 217. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-393. + 162. i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-354. - 354. i)T + 3.57e5iT^{2} \)
73 \( 1 + (22.2 - 22.2i)T - 3.89e5iT^{2} \)
79 \( 1 + 396. iT - 4.93e5T^{2} \)
83 \( 1 + (410. + 990. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-170. - 170. i)T + 7.04e5iT^{2} \)
97 \( 1 + 1.72e3T + 9.12e5T^{2} \)
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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

−11.65296342340808356401016157887, −10.87318908433895211894357174931, −9.365251674291305678192495585062, −8.876531291463069004838186755607, −7.70654799282350586419431493325, −6.94238954906098041496384325161, −5.57154426731199078233906595294, −4.11058068441926570830152865510, −2.82288231244314909283674436079, −1.58896293242729180070782838490, 0.989592823408894280714776959417, 2.90848256321028794873613558143, 3.86621120420895728833135877477, 5.16285229779735140041482330361, 6.48989279439248730551826414829, 7.88098286830896717284644368096, 8.569544503461126124078610766078, 9.375544211922749416842133442984, 10.52866303338187442616258712807, 11.33024191246383147222165778372

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