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Properties

Label 2-2e7-8.5-c3-0-9
Degree $2$
Conductor $128$
Sign $-0.707 + 0.707i$
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  − 8i·3-s − 12i·5-s + 32·7-s − 37·9-s − 8i·11-s − 20i·13-s − 96·15-s − 98·17-s + 88i·19-s − 256i·21-s − 32·23-s − 19·25-s + 80i·27-s + 172i·29-s + 256·31-s + ⋯
L(s)  = 1  − 1.53i·3-s − 1.07i·5-s + 1.72·7-s − 1.37·9-s − 0.219i·11-s − 0.426i·13-s − 1.65·15-s − 1.39·17-s + 1.06i·19-s − 2.66i·21-s − 0.290·23-s − 0.151·25-s + 0.570i·27-s + 1.10i·29-s + 1.48·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.707 + 0.707i$
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: \((2,\ 128,\ (\ :3/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.672513 - 1.62359i\)
\(L(\frac12)\) \(\approx\) \(0.672513 - 1.62359i\)
\(L(\frac{5}{2})\) 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 \( 1 + 8iT - 27T^{2} \)
5 \( 1 + 12iT - 125T^{2} \)
7 \( 1 - 32T + 343T^{2} \)
11 \( 1 + 8iT - 1.33e3T^{2} \)
13 \( 1 + 20iT - 2.19e3T^{2} \)
17 \( 1 + 98T + 4.91e3T^{2} \)
19 \( 1 - 88iT - 6.85e3T^{2} \)
23 \( 1 + 32T + 1.21e4T^{2} \)
29 \( 1 - 172iT - 2.43e4T^{2} \)
31 \( 1 - 256T + 2.97e4T^{2} \)
37 \( 1 + 92iT - 5.06e4T^{2} \)
41 \( 1 + 102T + 6.89e4T^{2} \)
43 \( 1 + 296iT - 7.95e4T^{2} \)
47 \( 1 - 320T + 1.03e5T^{2} \)
53 \( 1 + 76iT - 1.48e5T^{2} \)
59 \( 1 - 408iT - 2.05e5T^{2} \)
61 \( 1 - 636iT - 2.26e5T^{2} \)
67 \( 1 + 552iT - 3.00e5T^{2} \)
71 \( 1 - 416T + 3.57e5T^{2} \)
73 \( 1 + 138T + 3.89e5T^{2} \)
79 \( 1 - 64T + 4.93e5T^{2} \)
83 \( 1 + 392iT - 5.71e5T^{2} \)
89 \( 1 - 582T + 7.04e5T^{2} \)
97 \( 1 - 238T + 9.12e5T^{2} \)
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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

−12.42089260740481076117081928986, −11.83041443380580924483506319650, −10.70855406300588520073910657282, −8.679873894342876700325528142585, −8.255874575714206297938088971032, −7.20768799973568294427327842474, −5.75525133058376162544666269319, −4.58601960856081079575403047043, −2.01909209413364677560323945833, −0.987822623787148991049122495485, 2.48725669430463390720141254769, 4.23644807004496027662100866424, 4.92132557573469814446618530506, 6.62205745988845873508948989686, 8.089192463011981164000115636422, 9.191583735928561591017020860505, 10.33437900698030338384198933431, 11.12822947956863285704652408164, 11.57583110313415464142731708735, 13.67822346935608087163089189263

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