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Properties

Label 2-768-256.173-c1-0-17
Degree $2$
Conductor $768$
Sign $-0.938 - 0.345i$
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  + (1.19 + 0.754i)2-s + (0.514 + 0.857i)3-s + (0.862 + 1.80i)4-s + (−0.221 + 0.618i)5-s + (−0.0319 + 1.41i)6-s + (−3.13 − 0.949i)7-s + (−0.329 + 2.80i)8-s + (−0.471 + 0.881i)9-s + (−0.730 + 0.572i)10-s + (−4.69 + 3.48i)11-s + (−1.10 + 1.66i)12-s + (0.613 − 1.29i)13-s + (−3.02 − 3.49i)14-s + (−0.643 + 0.128i)15-s + (−2.51 + 3.11i)16-s + (−0.486 − 0.0967i)17-s + ⋯
L(s)  = 1  + (0.845 + 0.533i)2-s + (0.296 + 0.495i)3-s + (0.431 + 0.902i)4-s + (−0.0988 + 0.276i)5-s + (−0.0130 + 0.577i)6-s + (−1.18 − 0.358i)7-s + (−0.116 + 0.993i)8-s + (−0.157 + 0.293i)9-s + (−0.231 + 0.181i)10-s + (−1.41 + 1.04i)11-s + (−0.318 + 0.481i)12-s + (0.170 − 0.359i)13-s + (−0.809 − 0.934i)14-s + (−0.166 + 0.0330i)15-s + (−0.628 + 0.777i)16-s + (−0.118 − 0.0234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.938 - 0.345i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327398 + 1.83814i\)
\(L(\frac12)\) \(\approx\) \(0.327398 + 1.83814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.754i)T \)
3 \( 1 + (-0.514 - 0.857i)T \)
good5 \( 1 + (0.221 - 0.618i)T + (-3.86 - 3.17i)T^{2} \)
7 \( 1 + (3.13 + 0.949i)T + (5.82 + 3.88i)T^{2} \)
11 \( 1 + (4.69 - 3.48i)T + (3.19 - 10.5i)T^{2} \)
13 \( 1 + (-0.613 + 1.29i)T + (-8.24 - 10.0i)T^{2} \)
17 \( 1 + (0.486 + 0.0967i)T + (15.7 + 6.50i)T^{2} \)
19 \( 1 + (-0.963 - 0.0473i)T + (18.9 + 1.86i)T^{2} \)
23 \( 1 + (0.476 - 4.83i)T + (-22.5 - 4.48i)T^{2} \)
29 \( 1 + (-6.96 + 1.03i)T + (27.7 - 8.41i)T^{2} \)
31 \( 1 + (-3.03 + 1.25i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.220 + 0.199i)T + (3.62 + 36.8i)T^{2} \)
41 \( 1 + (2.56 - 2.10i)T + (7.99 - 40.2i)T^{2} \)
43 \( 1 + (-9.19 - 5.51i)T + (20.2 + 37.9i)T^{2} \)
47 \( 1 + (-2.81 + 1.87i)T + (17.9 - 43.4i)T^{2} \)
53 \( 1 + (-9.36 - 1.38i)T + (50.7 + 15.3i)T^{2} \)
59 \( 1 + (0.175 + 0.370i)T + (-37.4 + 45.6i)T^{2} \)
61 \( 1 + (-1.75 + 7.01i)T + (-53.7 - 28.7i)T^{2} \)
67 \( 1 + (8.94 + 2.23i)T + (59.0 + 31.5i)T^{2} \)
71 \( 1 + (6.98 - 3.73i)T + (39.4 - 59.0i)T^{2} \)
73 \( 1 + (8.60 - 2.60i)T + (60.6 - 40.5i)T^{2} \)
79 \( 1 + (-0.802 + 1.20i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (-7.17 + 6.49i)T + (8.13 - 82.6i)T^{2} \)
89 \( 1 + (0.415 + 4.21i)T + (-87.2 + 17.3i)T^{2} \)
97 \( 1 + (1.08 + 2.63i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.51843009674764024809703892805, −10.02263438161725072830728978049, −8.953533302934685158539466805181, −7.79494078292135219882303799434, −7.26137244593645555287949048160, −6.26584437517238200274803742785, −5.26909804054105661751784455181, −4.38476120602983610600713465796, −3.28352377481979925817726568791, −2.60935885712691848926657081516, 0.65731536979733053459558374085, 2.56143556127573719885482712249, 3.04909008142351273473995498586, 4.34864721518641686745416804299, 5.52515987165932203367239954745, 6.25236625266792939410699392255, 7.07668254874929614617434260417, 8.381259252171501194496704513563, 9.059213076163547240371815254844, 10.27725618719357787086202134378

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