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Properties

Label 2-768-256.165-c1-0-11
Degree $2$
Conductor $768$
Sign $0.935 - 0.352i$
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  + (0.574 − 1.29i)2-s + (−0.857 − 0.514i)3-s + (−1.34 − 1.48i)4-s + (−2.34 + 0.840i)5-s + (−1.15 + 0.813i)6-s + (−1.05 + 0.319i)7-s + (−2.68 + 0.879i)8-s + (0.471 + 0.881i)9-s + (−0.262 + 3.51i)10-s + (1.07 − 1.44i)11-s + (0.386 + 1.96i)12-s + (−2.39 + 1.13i)13-s + (−0.191 + 1.54i)14-s + (2.44 + 0.486i)15-s + (−0.407 + 3.97i)16-s + (6.36 − 1.26i)17-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.495 − 0.296i)3-s + (−0.670 − 0.742i)4-s + (−1.04 + 0.375i)5-s + (−0.472 + 0.331i)6-s + (−0.397 + 0.120i)7-s + (−0.950 + 0.310i)8-s + (0.157 + 0.293i)9-s + (−0.0831 + 1.11i)10-s + (0.323 − 0.435i)11-s + (0.111 + 0.566i)12-s + (−0.665 + 0.314i)13-s + (−0.0512 + 0.412i)14-s + (0.631 + 0.125i)15-s + (−0.101 + 0.994i)16-s + (1.54 − 0.307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.935 - 0.352i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678807 + 0.123582i\)
\(L(\frac12)\) \(\approx\) \(0.678807 + 0.123582i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.574 + 1.29i)T \)
3 \( 1 + (0.857 + 0.514i)T \)
good5 \( 1 + (2.34 - 0.840i)T + (3.86 - 3.17i)T^{2} \)
7 \( 1 + (1.05 - 0.319i)T + (5.82 - 3.88i)T^{2} \)
11 \( 1 + (-1.07 + 1.44i)T + (-3.19 - 10.5i)T^{2} \)
13 \( 1 + (2.39 - 1.13i)T + (8.24 - 10.0i)T^{2} \)
17 \( 1 + (-6.36 + 1.26i)T + (15.7 - 6.50i)T^{2} \)
19 \( 1 + (-0.0832 - 1.69i)T + (-18.9 + 1.86i)T^{2} \)
23 \( 1 + (-0.274 - 2.78i)T + (-22.5 + 4.48i)T^{2} \)
29 \( 1 + (1.38 - 9.34i)T + (-27.7 - 8.41i)T^{2} \)
31 \( 1 + (-6.35 - 2.63i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.88 - 2.08i)T + (-3.62 + 36.8i)T^{2} \)
41 \( 1 + (-1.68 - 1.37i)T + (7.99 + 40.2i)T^{2} \)
43 \( 1 + (5.00 + 8.35i)T + (-20.2 + 37.9i)T^{2} \)
47 \( 1 + (3.93 + 2.62i)T + (17.9 + 43.4i)T^{2} \)
53 \( 1 + (-0.252 - 1.70i)T + (-50.7 + 15.3i)T^{2} \)
59 \( 1 + (-5.35 - 2.53i)T + (37.4 + 45.6i)T^{2} \)
61 \( 1 + (13.3 - 3.33i)T + (53.7 - 28.7i)T^{2} \)
67 \( 1 + (-3.17 - 12.6i)T + (-59.0 + 31.5i)T^{2} \)
71 \( 1 + (-8.36 - 4.47i)T + (39.4 + 59.0i)T^{2} \)
73 \( 1 + (14.2 + 4.33i)T + (60.6 + 40.5i)T^{2} \)
79 \( 1 + (3.29 + 4.93i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-0.292 + 0.323i)T + (-8.13 - 82.6i)T^{2} \)
89 \( 1 + (0.335 - 3.40i)T + (-87.2 - 17.3i)T^{2} \)
97 \( 1 + (-0.969 + 2.34i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.49012225870764762423332933416, −9.846926068398299639703575111782, −8.797257923569223219087966297401, −7.73246889111135105189413401349, −6.84438803231885808499751360632, −5.75585247190965002179285793609, −4.87874286902503355762916144299, −3.67676961999083732613656580948, −2.99752828133127525355902245118, −1.26734982832766806635412275477, 0.37137596481496980309463429325, 3.12185700720178874010165704492, 4.17372714938252898603356811946, 4.77497506312712922858214325418, 5.87227183321584385142891166968, 6.70300767435826767562841854624, 7.80068955776514815058574576126, 8.107127148509647537763074264500, 9.509201920378509792966473337952, 9.972836174753024596099422317765

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