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Properties

Label 2-768-8.5-c1-0-7
Degree $2$
Conductor $768$
Sign $0.707 + 0.707i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
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  − i·3-s + 2i·5-s − 9-s − 4i·11-s − 2i·13-s + 2·15-s + 2·17-s − 4i·19-s + 8·23-s + 25-s + i·27-s + 6i·29-s + 8·31-s − 4·33-s − 6i·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.894i·5-s − 0.333·9-s − 1.20i·11-s − 0.554i·13-s + 0.516·15-s + 0.485·17-s − 0.917i·19-s + 1.66·23-s + 0.200·25-s + 0.192i·27-s + 1.11i·29-s + 1.43·31-s − 0.696·33-s − 0.986i·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40881 - 0.583548i\)
\(L(\frac12)\) \(\approx\) \(1.40881 - 0.583548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
good5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.53678054538526819429902130252, −9.221881939788189024996817046142, −8.488160237866709217936228452879, −7.48771610208422705976173381661, −6.79040057983196740369841487635, −5.94958079517156289184029044613, −4.95170658338410290830759369963, −3.29429813718658719386393794195, −2.75410683407352004343693392806, −0.912968318039718484977089011385, 1.34453381360352526103933658464, 2.89013753384545291997686266012, 4.34684880364153666893216696261, 4.76249930341247046819410826981, 5.87174871793317313828757853650, 6.98073821428419389803845495304, 8.009929251330985480190217384238, 8.819304113030716990485867333560, 9.677381846222648336617337407700, 10.14143597954171516893615931924

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