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Properties

Label 2-768-16.11-c2-0-4
Degree $2$
Conductor $768$
Sign $-0.130 - 0.991i$
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  + (−1.22 − 1.22i)3-s + (−3.71 − 3.71i)5-s + 5.63·7-s + 2.99i·9-s + (−5.26 + 5.26i)11-s + (−1.74 + 1.74i)13-s + 9.11i·15-s − 1.43·17-s + (−22.1 − 22.1i)19-s + (−6.90 − 6.90i)21-s + 2.27·23-s + 2.67i·25-s + (3.67 − 3.67i)27-s + (−35.9 + 35.9i)29-s − 13.3i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.743 − 0.743i)5-s + 0.805·7-s + 0.333i·9-s + (−0.478 + 0.478i)11-s + (−0.133 + 0.133i)13-s + 0.607i·15-s − 0.0846·17-s + (−1.16 − 1.16i)19-s + (−0.328 − 0.328i)21-s + 0.0988·23-s + 0.106i·25-s + (0.136 − 0.136i)27-s + (−1.23 + 1.23i)29-s − 0.429i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4437832653\)
\(L(\frac12)\) \(\approx\) \(0.4437832653\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (3.71 + 3.71i)T + 25iT^{2} \)
7 \( 1 - 5.63T + 49T^{2} \)
11 \( 1 + (5.26 - 5.26i)T - 121iT^{2} \)
13 \( 1 + (1.74 - 1.74i)T - 169iT^{2} \)
17 \( 1 + 1.43T + 289T^{2} \)
19 \( 1 + (22.1 + 22.1i)T + 361iT^{2} \)
23 \( 1 - 2.27T + 529T^{2} \)
29 \( 1 + (35.9 - 35.9i)T - 841iT^{2} \)
31 \( 1 + 13.3iT - 961T^{2} \)
37 \( 1 + (-36.1 - 36.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 63.2iT - 1.68e3T^{2} \)
43 \( 1 + (-46.5 + 46.5i)T - 1.84e3iT^{2} \)
47 \( 1 - 61.0iT - 2.20e3T^{2} \)
53 \( 1 + (-49.0 - 49.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (41.6 - 41.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (40.4 - 40.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (36.0 + 36.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 8.24T + 5.04e3T^{2} \)
73 \( 1 - 122. iT - 5.32e3T^{2} \)
79 \( 1 - 93.0iT - 6.24e3T^{2} \)
83 \( 1 + (21.6 + 21.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + 115.T + 9.40e3T^{2} \)
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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

−10.63069243278272995899805362746, −9.338548833131578365299324200779, −8.524090115326131670647475451430, −7.76382200302199982657100217764, −7.05842585875875206911682219734, −5.87196551623074335872722629611, −4.75024809738268473841348855919, −4.35348027650849886161940780595, −2.56808393544187156789346144326, −1.24146790078814760833780461837, 0.17150545487127535602635035360, 2.11102436211855810709529736861, 3.52715336159928936263680116840, 4.28947984537327231208700227450, 5.44619790066462470951834736570, 6.26194649295552488837572774304, 7.48522077447582607771321442391, 7.999508904288193224753278254599, 9.013573810061192210304367488309, 10.15639931547651782648539883888

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