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Properties

Label 2-768-32.29-c1-0-15
Degree $2$
Conductor $768$
Sign $-0.694 + 0.719i$
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  + (−0.923 − 0.382i)3-s + (−1.36 − 3.28i)5-s + (2.73 − 2.73i)7-s + (0.707 + 0.707i)9-s + (3.01 − 1.24i)11-s + (−0.932 + 2.25i)13-s + 3.55i·15-s + 0.517i·17-s + (1.52 − 3.68i)19-s + (−3.56 + 1.47i)21-s + (−2.39 − 2.39i)23-s + (−5.42 + 5.42i)25-s + (−0.382 − 0.923i)27-s + (−7.09 − 2.93i)29-s + 1.50·31-s + ⋯
L(s)  = 1  + (−0.533 − 0.220i)3-s + (−0.609 − 1.47i)5-s + (1.03 − 1.03i)7-s + (0.235 + 0.235i)9-s + (0.909 − 0.376i)11-s + (−0.258 + 0.624i)13-s + 0.918i·15-s + 0.125i·17-s + (0.350 − 0.845i)19-s + (−0.778 + 0.322i)21-s + (−0.500 − 0.500i)23-s + (−1.08 + 1.08i)25-s + (−0.0736 − 0.177i)27-s + (−1.31 − 0.545i)29-s + 0.269·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.694 + 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445932 - 1.05052i\)
\(L(\frac12)\) \(\approx\) \(0.445932 - 1.05052i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (1.36 + 3.28i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.73 + 2.73i)T - 7iT^{2} \)
11 \( 1 + (-3.01 + 1.24i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.932 - 2.25i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.517iT - 17T^{2} \)
19 \( 1 + (-1.52 + 3.68i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.39 + 2.39i)T + 23iT^{2} \)
29 \( 1 + (7.09 + 2.93i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 + (-3.40 - 8.22i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.21 + 3.21i)T + 41iT^{2} \)
43 \( 1 + (1.31 - 0.544i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.67iT - 47T^{2} \)
53 \( 1 + (-4.19 + 1.73i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.680 + 1.64i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (6.71 + 2.78i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-11.1 - 4.61i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (1.86 - 1.86i)T - 71iT^{2} \)
73 \( 1 + (9.06 + 9.06i)T + 73iT^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + (1.89 - 4.57i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.70 - 2.70i)T - 89iT^{2} \)
97 \( 1 - 3.73T + 97T^{2} \)
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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

−9.991157253287839689918781921393, −9.016883983614099288745950983005, −8.261357334692436795667683584903, −7.50710849577388859268631810900, −6.57667501434694725980415542920, −5.28823178528105089776128392215, −4.52256714219098543160136243598, −3.93822077972330768605519936991, −1.64794408956920822777374972753, −0.65964683598181297109073750080, 1.88764076234215303550287385731, 3.20368101071061984144204588772, 4.20630008912478666933473488981, 5.43866055738129540308456354881, 6.15676388561656550675686648113, 7.28790109170849471535478850732, 7.82613105628486364194310583264, 8.998923991824098340145926398946, 9.955267116304015488602110683175, 10.78149193772501815554033292496

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