Properties

Label 2-768-8.5-c3-0-10
Degree $2$
Conductor $768$
Sign $-0.707 - 0.707i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
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  − 3i·3-s + 18i·5-s + 8·7-s − 9·9-s + 36i·11-s − 10i·13-s + 54·15-s + 18·17-s + 100i·19-s − 24i·21-s + 72·23-s − 199·25-s + 27i·27-s − 234i·29-s + 16·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.60i·5-s + 0.431·7-s − 0.333·9-s + 0.986i·11-s − 0.213i·13-s + 0.929·15-s + 0.256·17-s + 1.20i·19-s − 0.249i·21-s + 0.652·23-s − 1.59·25-s + 0.192i·27-s − 1.49i·29-s + 0.0926·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.365251334\)
\(L(\frac12)\) \(\approx\) \(1.365251334\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3iT \)
good5 \( 1 - 18iT - 125T^{2} \)
7 \( 1 - 8T + 343T^{2} \)
11 \( 1 - 36iT - 1.33e3T^{2} \)
13 \( 1 + 10iT - 2.19e3T^{2} \)
17 \( 1 - 18T + 4.91e3T^{2} \)
19 \( 1 - 100iT - 6.85e3T^{2} \)
23 \( 1 - 72T + 1.21e4T^{2} \)
29 \( 1 + 234iT - 2.43e4T^{2} \)
31 \( 1 - 16T + 2.97e4T^{2} \)
37 \( 1 - 226iT - 5.06e4T^{2} \)
41 \( 1 + 90T + 6.89e4T^{2} \)
43 \( 1 - 452iT - 7.95e4T^{2} \)
47 \( 1 + 432T + 1.03e5T^{2} \)
53 \( 1 + 414iT - 1.48e5T^{2} \)
59 \( 1 + 684iT - 2.05e5T^{2} \)
61 \( 1 - 422iT - 2.26e5T^{2} \)
67 \( 1 + 332iT - 3.00e5T^{2} \)
71 \( 1 + 360T + 3.57e5T^{2} \)
73 \( 1 + 26T + 3.89e5T^{2} \)
79 \( 1 + 512T + 4.93e5T^{2} \)
83 \( 1 - 1.18e3iT - 5.71e5T^{2} \)
89 \( 1 - 630T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−10.14858071133076545281746931845, −9.739919847427455563574475338619, −8.132352628961608129278890661530, −7.69316273785688947708914519294, −6.72748946652048656650596668711, −6.17970192409433000064031607951, −4.90002723888209589110040794150, −3.57826537335481845767631838800, −2.60245583633063442516468550735, −1.57668740369942432676037242689, 0.36974274430274173369867956676, 1.47459208676132857585444529943, 3.11615049532934010268840929431, 4.31241149195333604815098441241, 5.06331162241425501708168017139, 5.67790826027805101546871684544, 7.05880997640315337451379325834, 8.247958534128808148070462846459, 8.878101281256257653792667858166, 9.271255407467152516286926728112

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