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Properties

Label 2-768-192.131-c1-0-9
Degree $2$
Conductor $768$
Sign $-0.0954 - 0.995i$
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  + (1.67 − 0.457i)3-s + (−1.45 + 2.18i)5-s + (1.27 + 3.08i)7-s + (2.58 − 1.52i)9-s + (−2.31 + 0.460i)11-s + (−5.55 + 3.71i)13-s + (−1.43 + 4.31i)15-s + (−3.42 + 3.42i)17-s + (1.23 − 0.822i)19-s + (3.54 + 4.56i)21-s + (1.87 − 4.52i)23-s + (−0.722 − 1.74i)25-s + (3.61 − 3.73i)27-s + (−0.760 + 3.82i)29-s + 0.0497·31-s + ⋯
L(s)  = 1  + (0.964 − 0.263i)3-s + (−0.652 + 0.975i)5-s + (0.482 + 1.16i)7-s + (0.860 − 0.509i)9-s + (−0.698 + 0.138i)11-s + (−1.54 + 1.02i)13-s + (−0.371 + 1.11i)15-s + (−0.829 + 0.829i)17-s + (0.282 − 0.188i)19-s + (0.773 + 0.997i)21-s + (0.391 − 0.944i)23-s + (−0.144 − 0.349i)25-s + (0.695 − 0.718i)27-s + (−0.141 + 0.710i)29-s + 0.00893·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0954 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.0954 - 0.995i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.0954 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08128 + 1.18993i\)
\(L(\frac12)\) \(\approx\) \(1.08128 + 1.18993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.457i)T \)
good5 \( 1 + (1.45 - 2.18i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-1.27 - 3.08i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.31 - 0.460i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (5.55 - 3.71i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (3.42 - 3.42i)T - 17iT^{2} \)
19 \( 1 + (-1.23 + 0.822i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-1.87 + 4.52i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.760 - 3.82i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 0.0497T + 31T^{2} \)
37 \( 1 + (-5.53 + 8.28i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-2.28 - 0.946i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.522 + 0.103i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (-2.51 - 2.51i)T + 47iT^{2} \)
53 \( 1 + (-1.79 - 9.00i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (2.56 + 1.71i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (0.935 - 4.70i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-3.65 - 0.726i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-12.6 + 5.23i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-13.6 - 5.64i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.69 - 1.69i)T + 79iT^{2} \)
83 \( 1 + (-5.63 - 8.42i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-8.26 + 3.42i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 3.93iT - 97T^{2} \)
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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

−10.61220118413463234000077018980, −9.453469038482762900762140952451, −8.862905450676459599973171947956, −7.890148533799254927046382287331, −7.26140450559123794180827644660, −6.45542866528436397799184572579, −5.01827891540366327818665358859, −4.02520761566814011913990831915, −2.63609113827652447083350090650, −2.24566027297677646489050652110, 0.70816116850808310406643586029, 2.44512147643773371943412337804, 3.63332130188770979902132840497, 4.71270414227102738774224584470, 5.07121692312821781728996205805, 7.05531587934770074016254229289, 7.87326987699682762217986660502, 8.042350778088217065109473988707, 9.308221236865538030594346553537, 9.938539037682994245050166221547

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