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Properties

Label 2-768-3.2-c2-0-43
Degree $2$
Conductor $768$
Sign $0.881 + 0.471i$
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  + (2.64 + 1.41i)3-s − 5.65i·5-s + 4·7-s + (5 + 7.48i)9-s − 8.48i·11-s + 10.5·13-s + (8.00 − 14.9i)15-s + 14.9i·17-s − 5.29·19-s + (10.5 + 5.65i)21-s − 29.9i·23-s − 7.00·25-s + (2.64 + 26.8i)27-s − 16.9i·29-s + 4·31-s + ⋯
L(s)  = 1  + (0.881 + 0.471i)3-s − 1.13i·5-s + 0.571·7-s + (0.555 + 0.831i)9-s − 0.771i·11-s + 0.814·13-s + (0.533 − 0.997i)15-s + 0.880i·17-s − 0.278·19-s + (0.503 + 0.269i)21-s − 1.30i·23-s − 0.280·25-s + (0.0979 + 0.995i)27-s − 0.585i·29-s + 0.129·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.849224427\)
\(L(\frac12)\) \(\approx\) \(2.849224427\)
\(L(2)\) 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 \( 1 + (-2.64 - 1.41i)T \)
good5 \( 1 + 5.65iT - 25T^{2} \)
7 \( 1 - 4T + 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 - 10.5T + 169T^{2} \)
17 \( 1 - 14.9iT - 289T^{2} \)
19 \( 1 + 5.29T + 361T^{2} \)
23 \( 1 + 29.9iT - 529T^{2} \)
29 \( 1 + 16.9iT - 841T^{2} \)
31 \( 1 - 4T + 961T^{2} \)
37 \( 1 - 52.9T + 1.36e3T^{2} \)
41 \( 1 + 29.9iT - 1.68e3T^{2} \)
43 \( 1 + 5.29T + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 50.9iT - 2.80e3T^{2} \)
59 \( 1 + 48.0iT - 3.48e3T^{2} \)
61 \( 1 - 95.2T + 3.72e3T^{2} \)
67 \( 1 - 47.6T + 4.48e3T^{2} \)
71 \( 1 - 89.7iT - 5.04e3T^{2} \)
73 \( 1 - 6T + 5.32e3T^{2} \)
79 \( 1 + 124T + 6.24e3T^{2} \)
83 \( 1 + 2.82iT - 6.88e3T^{2} \)
89 \( 1 - 104. iT - 7.92e3T^{2} \)
97 \( 1 - 118T + 9.40e3T^{2} \)
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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

−9.954577105309954322782849678582, −8.944177262625405339840432765901, −8.379558591409256628348243355128, −8.032371422866175913501444429913, −6.49741959741611075128797145913, −5.39282001499653927473633203524, −4.45640400140106467124871703830, −3.71769348938364777753852340830, −2.29286587066600443685351474201, −1.01372184291647032389329540319, 1.41007863511449309301620003212, 2.56497221087536445687691213890, 3.44303437590024790278593847159, 4.57748419707115684093989836501, 6.00022444133876336051597924029, 6.97969467622314810166845736684, 7.49610837951539818775388200649, 8.366925223234296653550785024611, 9.355902212060688083708090244683, 10.05989591051622628015792015174

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