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Properties

Label 2-2e7-32.29-c1-0-0
Degree $2$
Conductor $128$
Sign $0.831 - 0.555i$
Analytic cond. $1.02208$
Root an. cond. $1.01098$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.292i)3-s + (1.12 + 2.70i)5-s + (−1 + i)7-s + (−1.70 − 1.70i)9-s + (4.12 − 1.70i)11-s + (0.292 − 0.707i)13-s + 2.24i·15-s − 2.82i·17-s + (−1.53 + 3.70i)19-s + (−1 + 0.414i)21-s + (−5.82 − 5.82i)23-s + (−2.53 + 2.53i)25-s + (−1.58 − 3.82i)27-s + (−3.12 − 1.29i)29-s + 4·31-s + ⋯
L(s)  = 1  + (0.408 + 0.169i)3-s + (0.501 + 1.21i)5-s + (−0.377 + 0.377i)7-s + (−0.569 − 0.569i)9-s + (1.24 − 0.514i)11-s + (0.0812 − 0.196i)13-s + 0.579i·15-s − 0.685i·17-s + (−0.352 + 0.850i)19-s + (−0.218 + 0.0903i)21-s + (−1.21 − 1.21i)23-s + (−0.507 + 0.507i)25-s + (−0.305 − 0.736i)27-s + (−0.579 − 0.240i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ 0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18169 + 0.358462i\)
\(L(\frac12)\) \(\approx\) \(1.18169 + 0.358462i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.292i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.12 - 2.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (-4.12 + 1.70i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.292 + 0.707i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (1.53 - 3.70i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.82 + 5.82i)T + 23iT^{2} \)
29 \( 1 + (3.12 + 1.29i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-0.292 - 0.707i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.171 + 0.171i)T + 41iT^{2} \)
43 \( 1 + (4.70 - 1.94i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 0.343iT - 47T^{2} \)
53 \( 1 + (1.12 - 0.464i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.87 - 4.53i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.70 - 0.707i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-5.53 - 2.29i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-5.82 + 5.82i)T - 71iT^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (1.87 - 4.53i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-8.65 + 8.65i)T - 89iT^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−13.84805050718731728550387410621, −12.29247438258467652042646067708, −11.40786906507853736395475925501, −10.20561764554483086115359908040, −9.345175935594503143631372857265, −8.251659939268628398375211144370, −6.58073189263355946620158030198, −6.02892857622969747385174629820, −3.78317858452596858180933677268, −2.62267687397708442152110336678, 1.78087375666559412677570346116, 3.92492067149424886385048659598, 5.30527144455619649660238974294, 6.65067413958643854251610485494, 8.102120985609435411616459689078, 9.046186220322066091333800194428, 9.834339661981159416906086060811, 11.32535180546047285929218046225, 12.40579135043434849923329866182, 13.35539447596603023025222145429

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