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Properties

Label 2-2e7-32.21-c3-0-6
Degree $2$
Conductor $128$
Sign $-0.377 + 0.925i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.377 + 0.925i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.377 + 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.330846 - 0.492280i\)
\(L(\frac12)\) \(\approx\) \(0.330846 - 0.492280i\)
\(L(\frac{5}{2})\) 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

−12.53251607993802953894290045473, −11.28934631851355965461296544262, −10.76048912238801406044457461879, −9.481166495216801740707874089291, −8.317214510406595350543854938445, −6.98187386513565495628708789177, −5.32961020923782425887239685631, −5.01281655259034038182286487696, −2.76869368660602743888250225825, −0.32971439258420958348719613579, 1.80181015784927710257478356730, 3.98937379322205847613394840151, 5.50365988114107593685169211216, 6.53541919995726997465115551929, 7.58165039129959325128867825200, 8.991451152454276245847192273843, 10.36957309954482377665589743541, 11.13331040201710113038909745720, 12.17018766478191320073743234161, 12.91929206337215379799019618950

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