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Properties

Label 2-768-32.29-c1-0-6
Degree $2$
Conductor $768$
Sign $0.985 - 0.172i$
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 + (0.155 + 0.375i)5-s + (0.709 − 0.709i)7-s + (0.707 + 0.707i)9-s + (2.79 − 1.15i)11-s + (−2.58 + 6.24i)13-s − 0.406i·15-s + 1.05i·17-s + (1.48 − 3.59i)19-s + (−0.926 + 0.383i)21-s + (0.922 + 0.922i)23-s + (3.41 − 3.41i)25-s + (−0.382 − 0.923i)27-s + (7.64 + 3.16i)29-s − 1.88·31-s + ⋯
L(s)  = 1  + (−0.533 − 0.220i)3-s + (0.0696 + 0.168i)5-s + (0.268 − 0.268i)7-s + (0.235 + 0.235i)9-s + (0.841 − 0.348i)11-s + (−0.717 + 1.73i)13-s − 0.105i·15-s + 0.256i·17-s + (0.341 − 0.823i)19-s + (−0.202 + 0.0837i)21-s + (0.192 + 0.192i)23-s + (0.683 − 0.683i)25-s + (−0.0736 − 0.177i)27-s + (1.41 + 0.587i)29-s − 0.338·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.985 - 0.172i$
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: \((2,\ 768,\ (\ :1/2),\ 0.985 - 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37184 + 0.118917i\)
\(L(\frac12)\) \(\approx\) \(1.37184 + 0.118917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.155 - 0.375i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.709 + 0.709i)T - 7iT^{2} \)
11 \( 1 + (-2.79 + 1.15i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.58 - 6.24i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 1.05iT - 17T^{2} \)
19 \( 1 + (-1.48 + 3.59i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.922 - 0.922i)T + 23iT^{2} \)
29 \( 1 + (-7.64 - 3.16i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.88T + 31T^{2} \)
37 \( 1 + (1.24 + 3.01i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.11 - 5.11i)T + 41iT^{2} \)
43 \( 1 + (-10.9 + 4.53i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 7.47iT - 47T^{2} \)
53 \( 1 + (-7.58 + 3.14i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-4.13 - 9.98i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.35 + 0.562i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (10.8 + 4.50i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-9.35 + 9.35i)T - 71iT^{2} \)
73 \( 1 + (-0.367 - 0.367i)T + 73iT^{2} \)
79 \( 1 + 5.87iT - 79T^{2} \)
83 \( 1 + (1.62 - 3.91i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-8.33 + 8.33i)T - 89iT^{2} \)
97 \( 1 + 7.97T + 97T^{2} \)
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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

−10.54236021591503363467998419623, −9.363958382474051992084805619246, −8.855575061657207717410250099392, −7.50875608825585301558146197160, −6.82606259060133317512885396998, −6.12295902600446759455920046828, −4.82091890996554140979451164513, −4.14319922698203566361825292054, −2.57081986864185707591790168646, −1.16589151008672316964026115705, 0.968089744250196367106200226359, 2.65707209754993718739635616666, 3.93687777797677091606417906982, 5.07999110699398112211561131412, 5.66236647602624848316176677546, 6.78782631273619560392562339065, 7.70201741320640276752208232060, 8.615839330262723936955709185093, 9.616039580441616581289900902282, 10.26784862178551804554959301282

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