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Properties

Label 2-56-56.51-c2-0-9
Degree $2$
Conductor $56$
Sign $0.357 + 0.933i$
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  + (1.51 − 1.30i)2-s + (−0.824 − 1.42i)3-s + (0.597 − 3.95i)4-s + (3.95 + 2.28i)5-s + (−3.11 − 1.08i)6-s + (−6.75 + 1.83i)7-s + (−4.25 − 6.77i)8-s + (3.14 − 5.43i)9-s + (8.96 − 1.69i)10-s + (6.18 + 10.7i)11-s + (−6.13 + 2.40i)12-s + 18.3i·13-s + (−7.84 + 11.5i)14-s − 7.52i·15-s + (−15.2 − 4.72i)16-s + (6.51 + 11.2i)17-s + ⋯
L(s)  = 1  + (0.758 − 0.652i)2-s + (−0.274 − 0.475i)3-s + (0.149 − 0.988i)4-s + (0.790 + 0.456i)5-s + (−0.518 − 0.181i)6-s + (−0.965 + 0.262i)7-s + (−0.531 − 0.846i)8-s + (0.348 − 0.604i)9-s + (0.896 − 0.169i)10-s + (0.562 + 0.974i)11-s + (−0.511 + 0.200i)12-s + 1.41i·13-s + (−0.560 + 0.828i)14-s − 0.501i·15-s + (−0.955 − 0.295i)16-s + (0.383 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.357 + 0.933i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.357 + 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.32029 - 0.908489i\)
\(L(\frac12)\) \(\approx\) \(1.32029 - 0.908489i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.51 + 1.30i)T \)
7 \( 1 + (6.75 - 1.83i)T \)
good3 \( 1 + (0.824 + 1.42i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.95 - 2.28i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.18 - 10.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 18.3iT - 169T^{2} \)
17 \( 1 + (-6.51 - 11.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.51 + 2.61i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (26.2 + 15.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 22.7iT - 841T^{2} \)
31 \( 1 + (-19.5 + 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.9 - 6.88i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 60.5T + 1.68e3T^{2} \)
43 \( 1 - 39.0T + 1.84e3T^{2} \)
47 \( 1 + (17.6 + 10.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-4.12 + 2.38i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (5.86 + 10.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (94.3 + 54.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.5 - 68.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 12.9iT - 5.04e3T^{2} \)
73 \( 1 + (-49.2 - 85.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-113. - 65.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 28.3T + 6.88e3T^{2} \)
89 \( 1 + (-78.7 + 136. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 39.6T + 9.40e3T^{2} \)
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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

−14.51086625853022167099840410759, −13.56336261034981195373125549887, −12.44389383687430210931016962614, −11.80909200620076336367041660517, −10.06684149691467736920806786681, −9.489115937756688847661981419626, −6.67625262243649254163725610662, −6.19731219643968037565963964952, −4.06480540756194897900509583391, −2.04282147504284973437301999049, 3.41866432387190284203655925322, 5.20789638546566710324335634670, 6.13567208553304390812319195951, 7.78613370424043995459155607560, 9.357687295267018000919722863457, 10.57431561350234201179403227752, 12.16220526161617650006363317799, 13.33933039959611937075924384049, 13.84846093358628413265065274208, 15.44436640565502886013191327384

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