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Properties

Label 2-2e7-8.5-c3-0-5
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

\(L(2)\) \(\approx\) \(1.86733\)
\(L(\frac12)\) \(\approx\) \(1.86733\)
\(L(\frac{5}{2})\) 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

−12.81146615570764980042204002957, −11.70129097841085557184618340537, −10.82447452772290597494812629115, −9.582775670188518468128407765530, −8.730562439661298931194605571401, −7.934803100557612782905712947515, −5.57790411167408496988752528083, −4.87596700180645501866596888215, −3.79679415022851530481763655912, −1.20462889427626375237706648641, 1.58060051165689216341964464967, 2.91788851574558275541648851809, 5.00896723109477689334745653313, 6.74195622579686793005980169405, 7.22580022671621646331627904796, 8.163832432087228483075589236271, 9.949161790519663999598469526772, 10.99007053273707497041302588095, 11.80680442221122422453218854430, 12.87451666649714569753020263937

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