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Properties

Label 2-56-56.45-c2-0-3
Degree $2$
Conductor $56$
Sign $0.961 - 0.274i$
Analytic cond. $1.52588$
Root an. cond. $1.23526$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 1.18i)2-s + (−0.455 + 0.788i)3-s + (1.21 + 3.81i)4-s + (3.17 + 5.49i)5-s + (1.66 − 0.735i)6-s + (3.79 − 5.88i)7-s + (2.54 − 7.58i)8-s + (4.08 + 7.07i)9-s + (1.36 − 12.6i)10-s + (11.4 + 6.60i)11-s + (−3.55 − 0.780i)12-s − 19.4·13-s + (−13.0 + 5.02i)14-s − 5.77·15-s + (−13.0 + 9.23i)16-s + (13.7 + 7.96i)17-s + ⋯
L(s)  = 1  + (−0.807 − 0.590i)2-s + (−0.151 + 0.262i)3-s + (0.302 + 0.953i)4-s + (0.634 + 1.09i)5-s + (0.277 − 0.122i)6-s + (0.541 − 0.840i)7-s + (0.318 − 0.948i)8-s + (0.453 + 0.786i)9-s + (0.136 − 1.26i)10-s + (1.04 + 0.600i)11-s + (−0.296 − 0.0650i)12-s − 1.49·13-s + (−0.933 + 0.358i)14-s − 0.385·15-s + (−0.816 + 0.577i)16-s + (0.811 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.961 - 0.274i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.961 - 0.274i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.886465 + 0.123954i\)
\(L(\frac12)\) \(\approx\) \(0.886465 + 0.123954i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 + 1.18i)T \)
7 \( 1 + (-3.79 + 5.88i)T \)
good3 \( 1 + (0.455 - 0.788i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.17 - 5.49i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-11.4 - 6.60i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 19.4T + 169T^{2} \)
17 \( 1 + (-13.7 - 7.96i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8.22 + 14.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (11.9 + 20.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 16.6iT - 841T^{2} \)
31 \( 1 + (11.1 + 6.42i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-41.1 + 23.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 6.49iT - 1.68e3T^{2} \)
43 \( 1 + 33.2iT - 1.84e3T^{2} \)
47 \( 1 + (18.9 - 10.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (32.2 + 18.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (27.3 - 47.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.12 + 8.87i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (14.8 + 8.56i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 32.0T + 5.04e3T^{2} \)
73 \( 1 + (-92.8 - 53.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-29.1 - 50.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 36.3T + 6.88e3T^{2} \)
89 \( 1 + (-0.929 + 0.536i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 169. iT - 9.40e3T^{2} \)
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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

−14.94482984574470683480152916801, −14.04534197078901809271840830964, −12.58547182769300922157104367483, −11.23754965663035099093792768764, −10.31342309848329452162070327521, −9.692268941138056409732732889575, −7.74385348169240557305315874022, −6.81102569646091957716389781215, −4.32624975792848608787510687155, −2.19351991774115985338057099735, 1.45125959775547069828490827300, 5.11167455081702302960736419640, 6.22549118930153437407486094579, 7.83211655555182777770239555687, 9.154757102299015966986765488992, 9.705877053641583069847783157457, 11.68060324479783916468214482714, 12.50248611961897335095213069978, 14.19926688981375236006232674022, 15.01308636626623354525270528059

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