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Properties

Label 2-768-32.13-c1-0-15
Degree $2$
Conductor $768$
Sign $-0.987 + 0.154i$
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  + (0.382 − 0.923i)3-s + (−0.825 + 0.342i)5-s + (−1.17 + 1.17i)7-s + (−0.707 − 0.707i)9-s + (−1.46 − 3.53i)11-s + (−3.01 − 1.24i)13-s + 0.893i·15-s + 4.58i·17-s + (−3.29 − 1.36i)19-s + (0.637 + 1.53i)21-s + (−5.41 − 5.41i)23-s + (−2.97 + 2.97i)25-s + (−0.923 + 0.382i)27-s + (2.46 − 5.95i)29-s + 5.25·31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (−0.369 + 0.152i)5-s + (−0.445 + 0.445i)7-s + (−0.235 − 0.235i)9-s + (−0.441 − 1.06i)11-s + (−0.834 − 0.345i)13-s + 0.230i·15-s + 1.11i·17-s + (−0.757 − 0.313i)19-s + (0.139 + 0.335i)21-s + (−1.12 − 1.12i)23-s + (−0.594 + 0.594i)25-s + (−0.177 + 0.0736i)27-s + (0.457 − 1.10i)29-s + 0.943·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.987 + 0.154i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0298456 - 0.382858i\)
\(L(\frac12)\) \(\approx\) \(0.0298456 - 0.382858i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (0.825 - 0.342i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.17 - 1.17i)T - 7iT^{2} \)
11 \( 1 + (1.46 + 3.53i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (3.01 + 1.24i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.58iT - 17T^{2} \)
19 \( 1 + (3.29 + 1.36i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.41 + 5.41i)T + 23iT^{2} \)
29 \( 1 + (-2.46 + 5.95i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 5.25T + 31T^{2} \)
37 \( 1 + (7.33 - 3.03i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.35 + 1.35i)T + 41iT^{2} \)
43 \( 1 + (2.95 + 7.14i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 8.16iT - 47T^{2} \)
53 \( 1 + (-3.13 - 7.57i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.221 + 0.0919i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.66 - 6.44i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.52 - 13.3i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (1.51 - 1.51i)T - 71iT^{2} \)
73 \( 1 + (9.62 + 9.62i)T + 73iT^{2} \)
79 \( 1 + 5.34iT - 79T^{2} \)
83 \( 1 + (-5.64 - 2.33i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.09 + 5.09i)T - 89iT^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−10.11577927549858388127057679675, −8.715730220813074572506902927095, −8.336400458839273091497172493845, −7.39248650809990322214338103674, −6.32605627668242118327980843907, −5.72142732902137325936064371755, −4.30414477978616931051707433056, −3.15384036188845103244728871208, −2.18251216090276477206847365523, −0.17336485953180039342693193733, 2.10111808094429162845587119101, 3.36631057881568638129772174385, 4.43342106559727601091552885069, 5.07495296657603069983768619306, 6.46917430460209026606560769648, 7.38896155666655037618599604537, 8.071603701085999248399837689786, 9.234877187808694474210938315868, 9.928878035166079827516459528446, 10.40825841550404640219982005345

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