↑

Properties

Label 2-768-256.157-c1-0-20
Degree $2$
Conductor $768$
Sign $0.307 - 0.951i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.499 + 1.32i)2-s + (0.740 − 0.671i)3-s + (−1.50 − 1.32i)4-s + (−0.0500 − 0.199i)5-s + (0.518 + 1.31i)6-s + (3.64 + 2.98i)7-s + (2.49 − 1.32i)8-s + (0.0980 − 0.995i)9-s + (0.289 + 0.0336i)10-s + (0.514 − 0.243i)11-s + (−1.99 + 0.0275i)12-s + (−3.60 + 2.15i)13-s + (−5.77 + 3.32i)14-s + (−0.171 − 0.114i)15-s + (0.501 + 3.96i)16-s + (−2.22 + 1.48i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s + (0.427 − 0.387i)3-s + (−0.750 − 0.661i)4-s + (−0.0223 − 0.0894i)5-s + (0.211 + 0.537i)6-s + (1.37 + 1.12i)7-s + (0.883 − 0.467i)8-s + (0.0326 − 0.331i)9-s + (0.0915 + 0.0106i)10-s + (0.155 − 0.0733i)11-s + (−0.577 + 0.00795i)12-s + (−0.998 + 0.598i)13-s + (−1.54 + 0.888i)14-s + (−0.0442 − 0.0295i)15-s + (0.125 + 0.992i)16-s + (−0.538 + 0.359i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.307 - 0.951i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.307 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26282 + 0.918551i\)
\(L(\frac12)\) \(\approx\) \(1.26282 + 0.918551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.499 - 1.32i)T \)
3 \( 1 + (-0.740 + 0.671i)T \)
good5 \( 1 + (0.0500 + 0.199i)T + (-4.40 + 2.35i)T^{2} \)
7 \( 1 + (-3.64 - 2.98i)T + (1.36 + 6.86i)T^{2} \)
11 \( 1 + (-0.514 + 0.243i)T + (6.97 - 8.50i)T^{2} \)
13 \( 1 + (3.60 - 2.15i)T + (6.12 - 11.4i)T^{2} \)
17 \( 1 + (2.22 - 1.48i)T + (6.50 - 15.7i)T^{2} \)
19 \( 1 + (-0.651 + 0.878i)T + (-5.51 - 18.1i)T^{2} \)
23 \( 1 + (-7.58 + 2.30i)T + (19.1 - 12.7i)T^{2} \)
29 \( 1 + (-3.80 + 1.36i)T + (22.4 - 18.3i)T^{2} \)
31 \( 1 + (-2.06 - 4.98i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.07 - 7.22i)T + (-35.4 - 10.7i)T^{2} \)
41 \( 1 + (-2.22 - 1.18i)T + (22.7 + 34.0i)T^{2} \)
43 \( 1 + (3.99 - 4.40i)T + (-4.21 - 42.7i)T^{2} \)
47 \( 1 + (-0.896 + 4.50i)T + (-43.4 - 17.9i)T^{2} \)
53 \( 1 + (-3.70 - 1.32i)T + (40.9 + 33.6i)T^{2} \)
59 \( 1 + (9.62 + 5.76i)T + (27.8 + 52.0i)T^{2} \)
61 \( 1 + (1.59 + 0.0782i)T + (60.7 + 5.97i)T^{2} \)
67 \( 1 + (0.515 - 10.4i)T + (-66.6 - 6.56i)T^{2} \)
71 \( 1 + (-0.180 + 0.0177i)T + (69.6 - 13.8i)T^{2} \)
73 \( 1 + (-9.23 + 7.57i)T + (14.2 - 71.5i)T^{2} \)
79 \( 1 + (5.12 - 1.01i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-0.0740 - 0.499i)T + (-79.4 + 24.0i)T^{2} \)
89 \( 1 + (5.43 + 1.64i)T + (74.0 + 49.4i)T^{2} \)
97 \( 1 + (-13.7 + 5.68i)T + (68.5 - 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.30833528719327111302694383282, −9.142059520271709857317533961780, −8.687369948591493984232947574976, −8.067409537118584214482182400328, −7.05809752228182887192431092904, −6.33903300676732081102792200140, −4.98138326373042959353385504665, −4.68872315264098218792925614285, −2.66589200891282363670897528905, −1.40570246239696846559361640982, 1.02262461274875793951793055870, 2.38674256324258589635317598640, 3.52178314890659975913951890666, 4.61222478056184081985709558484, 5.07718363415343472056043029382, 7.23758172711711787826564481998, 7.62796749281494146889980970888, 8.632099016279289376751253477920, 9.380723750446062114339552085151, 10.33684958975493154625768680090

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