Properties

Label 2-768-24.5-c2-0-25
Degree $2$
Conductor $768$
Sign $0.884 + 0.465i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.888 + 2.86i)3-s − 8.59·5-s − 10.9·7-s + (−7.41 + 5.09i)9-s + 2.75·11-s − 4.43i·13-s + (−7.63 − 24.6i)15-s + 25.4i·17-s − 17.5i·19-s + (−9.71 − 31.3i)21-s + 17.5i·23-s + 48.8·25-s + (−21.1 − 16.7i)27-s + 19.6·29-s + 2.58·31-s + ⋯
L(s)  = 1  + (0.296 + 0.955i)3-s − 1.71·5-s − 1.56·7-s + (−0.824 + 0.565i)9-s + 0.250·11-s − 0.341i·13-s + (−0.509 − 1.64i)15-s + 1.49i·17-s − 0.923i·19-s + (−0.462 − 1.49i)21-s + 0.762i·23-s + 1.95·25-s + (−0.784 − 0.619i)27-s + 0.676·29-s + 0.0833·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.884 + 0.465i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.884 + 0.465i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5283659056\)
\(L(\frac12)\) \(\approx\) \(0.5283659056\)
\(L(2)\) 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 \( 1 + (-0.888 - 2.86i)T \)
good5 \( 1 + 8.59T + 25T^{2} \)
7 \( 1 + 10.9T + 49T^{2} \)
11 \( 1 - 2.75T + 121T^{2} \)
13 \( 1 + 4.43iT - 169T^{2} \)
17 \( 1 - 25.4iT - 289T^{2} \)
19 \( 1 + 17.5iT - 361T^{2} \)
23 \( 1 - 17.5iT - 529T^{2} \)
29 \( 1 - 19.6T + 841T^{2} \)
31 \( 1 - 2.58T + 961T^{2} \)
37 \( 1 + 7.73iT - 1.36e3T^{2} \)
41 \( 1 + 58.0iT - 1.68e3T^{2} \)
43 \( 1 + 42.1iT - 1.84e3T^{2} \)
47 \( 1 - 17.4iT - 2.20e3T^{2} \)
53 \( 1 - 69.0T + 2.80e3T^{2} \)
59 \( 1 + 50.5T + 3.48e3T^{2} \)
61 \( 1 + 32.5iT - 3.72e3T^{2} \)
67 \( 1 - 48.0iT - 4.48e3T^{2} \)
71 \( 1 - 22.1iT - 5.04e3T^{2} \)
73 \( 1 - 27.0T + 5.32e3T^{2} \)
79 \( 1 + 97.4T + 6.24e3T^{2} \)
83 \( 1 + 59.5T + 6.88e3T^{2} \)
89 \( 1 - 110. iT - 7.92e3T^{2} \)
97 \( 1 - 55.1T + 9.40e3T^{2} \)
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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

−10.15584016069659651212634129877, −9.093868695789340690525802241010, −8.526930822082158493061672412851, −7.55851207878465191023794581090, −6.64683763738751479743256329620, −5.49541265630460078015068066771, −4.15512749761863396974855858012, −3.70115058229325360840845735752, −2.86607456092132681781304266166, −0.26488492040350664953393411886, 0.77606310281353952712315403517, 2.80488177212714014680264936828, 3.44512436463441630251226068835, 4.53975858275202251518678512472, 6.14952658432031917630699150896, 6.86862575552302100263313032542, 7.50407224716532848877780321871, 8.355200136652196361247631654450, 9.156391346876990241628364269484, 10.09599755349002135957796380846

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