Properties

Label 2-2e7-8.5-c3-0-11
Degree $2$
Conductor $128$
Sign $-1$
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  − 6.32i·3-s − 17.8i·5-s − 22.6·7-s − 13.0·9-s + 44.2i·11-s + 17.8i·13-s − 113.·15-s + 70·17-s − 82.2i·19-s + 143. i·21-s − 158.·23-s − 195.·25-s − 88.5i·27-s − 125. i·29-s + 280·33-s + ⋯
L(s)  = 1  − 1.21i·3-s − 1.59i·5-s − 1.22·7-s − 0.481·9-s + 1.21i·11-s + 0.381i·13-s − 1.94·15-s + 0.998·17-s − 0.992i·19-s + 1.48i·21-s − 1.43·23-s − 1.56·25-s − 0.631i·27-s − 0.801i·29-s + 1.47·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-1$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(\approx\) \(-1.03176i\)
\(L(\frac12)\) \(\approx\) \(-1.03176i\)
\(L(\frac{5}{2})\) 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 \( 1 + 6.32iT - 27T^{2} \)
5 \( 1 + 17.8iT - 125T^{2} \)
7 \( 1 + 22.6T + 343T^{2} \)
11 \( 1 - 44.2iT - 1.33e3T^{2} \)
13 \( 1 - 17.8iT - 2.19e3T^{2} \)
17 \( 1 - 70T + 4.91e3T^{2} \)
19 \( 1 + 82.2iT - 6.85e3T^{2} \)
23 \( 1 + 158.T + 1.21e4T^{2} \)
29 \( 1 + 125. iT - 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 375. iT - 5.06e4T^{2} \)
41 \( 1 + 182T + 6.89e4T^{2} \)
43 \( 1 + 132. iT - 7.95e4T^{2} \)
47 \( 1 - 316.T + 1.03e5T^{2} \)
53 \( 1 - 125. iT - 1.48e5T^{2} \)
59 \( 1 + 82.2iT - 2.05e5T^{2} \)
61 \( 1 - 232. iT - 2.26e5T^{2} \)
67 \( 1 - 221. iT - 3.00e5T^{2} \)
71 \( 1 + 113.T + 3.57e5T^{2} \)
73 \( 1 - 910T + 3.89e5T^{2} \)
79 \( 1 - 678.T + 4.93e5T^{2} \)
83 \( 1 + 714. iT - 5.71e5T^{2} \)
89 \( 1 + 546T + 7.04e5T^{2} \)
97 \( 1 + 490T + 9.12e5T^{2} \)
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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

−12.40832890459561949129560088840, −12.03581223370086710500124360580, −9.947518204547707664068174667346, −9.179098137008116716528552416196, −7.912089215880290909214454444923, −6.93904359422186065305981708579, −5.71317941436571232451105364763, −4.22586096781788846847025218758, −2.01542653512916390805880529757, −0.52240111283131630807705731875, 3.15706713652802246004115415058, 3.62644832231848533663856288426, 5.70107624936722437144644396843, 6.61216925073114069213018639094, 8.082688529071269249181523519771, 9.694926080346705034738807088031, 10.20748072813757285961104199978, 10.93403934088402298490618874132, 12.19068465020443144312628544609, 13.67899429290759603254175071055

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