Properties

Label 2-768-96.35-c1-0-18
Degree $2$
Conductor $768$
Sign $0.0914 + 0.995i$
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  + (0.0380 + 1.73i)3-s + (−2.18 + 0.906i)5-s + (−1.93 − 1.93i)7-s + (−2.99 + 0.131i)9-s + (1.42 − 0.590i)11-s + (0.110 − 0.266i)13-s + (−1.65 − 3.75i)15-s + 6.17·17-s + (−7.34 − 3.04i)19-s + (3.27 − 3.41i)21-s + (−1.85 − 1.85i)23-s + (0.430 − 0.430i)25-s + (−0.342 − 5.18i)27-s + (2.11 − 5.10i)29-s − 3.42i·31-s + ⋯
L(s)  = 1  + (0.0219 + 0.999i)3-s + (−0.978 + 0.405i)5-s + (−0.730 − 0.730i)7-s + (−0.999 + 0.0439i)9-s + (0.429 − 0.177i)11-s + (0.0306 − 0.0739i)13-s + (−0.426 − 0.969i)15-s + 1.49·17-s + (−1.68 − 0.697i)19-s + (0.714 − 0.746i)21-s + (−0.386 − 0.386i)23-s + (0.0861 − 0.0861i)25-s + (−0.0658 − 0.997i)27-s + (0.392 − 0.948i)29-s − 0.614i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0914 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0914 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.0914 + 0.995i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.0914 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350171 - 0.319480i\)
\(L(\frac12)\) \(\approx\) \(0.350171 - 0.319480i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0380 - 1.73i)T \)
good5 \( 1 + (2.18 - 0.906i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.93 + 1.93i)T + 7iT^{2} \)
11 \( 1 + (-1.42 + 0.590i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.110 + 0.266i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.17T + 17T^{2} \)
19 \( 1 + (7.34 + 3.04i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.85 + 1.85i)T + 23iT^{2} \)
29 \( 1 + (-2.11 + 5.10i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 3.42iT - 31T^{2} \)
37 \( 1 + (2.52 + 6.09i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.753 + 0.753i)T - 41iT^{2} \)
43 \( 1 + (1.57 + 3.79i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.54iT - 47T^{2} \)
53 \( 1 + (-5.12 - 12.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.08 + 7.45i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-4.28 - 1.77i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-0.531 + 1.28i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (8.72 - 8.72i)T - 71iT^{2} \)
73 \( 1 + (2.73 + 2.73i)T + 73iT^{2} \)
79 \( 1 + 2.76T + 79T^{2} \)
83 \( 1 + (-2.53 + 6.11i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.14 + 4.14i)T + 89iT^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−10.26914109169317202788756554086, −9.396010736601441161012995344107, −8.444461127052169342617275081841, −7.61130790069541421491266257978, −6.61414151724553005591058501460, −5.66693477461219943439179556497, −4.20102647389206336333366648476, −3.87278468221386671035175459423, −2.79299446699565377064839186476, −0.24382731706087133603410348062, 1.48721161258338831908941780796, 2.95221169791383280906943698924, 3.94305726107804760263153966694, 5.34327230477302281001963302972, 6.27059062415339959058000622596, 7.02470561419321613673192268482, 8.155927304853292600979226644604, 8.454604612833070743314316323861, 9.541304007469445729695425783164, 10.57607612292603108548657713676

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