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Properties

Label 2-56-56.45-c2-0-11
Degree $2$
Conductor $56$
Sign $0.495 + 0.868i$
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  + (1.54 − 1.26i)2-s + (0.126 − 0.219i)3-s + (0.785 − 3.92i)4-s + (−1.78 − 3.09i)5-s + (−0.0821 − 0.499i)6-s + (2.89 + 6.37i)7-s + (−3.75 − 7.06i)8-s + (4.46 + 7.73i)9-s + (−6.68 − 2.52i)10-s + (6.82 + 3.94i)11-s + (−0.760 − 0.668i)12-s − 18.1·13-s + (12.5 + 6.18i)14-s − 0.904·15-s + (−14.7 − 6.16i)16-s + (−8.26 − 4.76i)17-s + ⋯
L(s)  = 1  + (0.773 − 0.633i)2-s + (0.0422 − 0.0731i)3-s + (0.196 − 0.980i)4-s + (−0.357 − 0.618i)5-s + (−0.0136 − 0.0833i)6-s + (0.413 + 0.910i)7-s + (−0.469 − 0.882i)8-s + (0.496 + 0.859i)9-s + (−0.668 − 0.252i)10-s + (0.620 + 0.358i)11-s + (−0.0633 − 0.0557i)12-s − 1.39·13-s + (0.896 + 0.442i)14-s − 0.0603·15-s + (−0.922 − 0.385i)16-s + (−0.485 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.495 + 0.868i$
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: \((2,\ 56,\ (\ :1),\ 0.495 + 0.868i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42821 - 0.829558i\)
\(L(\frac12)\) \(\approx\) \(1.42821 - 0.829558i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.54 + 1.26i)T \)
7 \( 1 + (-2.89 - 6.37i)T \)
good3 \( 1 + (-0.126 + 0.219i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.78 + 3.09i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.82 - 3.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 18.1T + 169T^{2} \)
17 \( 1 + (8.26 + 4.76i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-12.4 - 21.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.14 - 3.72i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 28.3iT - 841T^{2} \)
31 \( 1 + (28.2 + 16.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (25.9 - 14.9i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 45.2iT - 1.68e3T^{2} \)
43 \( 1 + 24.9iT - 1.84e3T^{2} \)
47 \( 1 + (-44.0 + 25.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-54.3 - 31.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (37.0 - 64.0i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25.2 - 43.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (108. + 62.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 5.33T + 5.04e3T^{2} \)
73 \( 1 + (23.6 + 13.6i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-51.5 - 89.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 51.5T + 6.88e3T^{2} \)
89 \( 1 + (-133. + 76.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 47.0iT - 9.40e3T^{2} \)
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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

−14.72723235721723345884811575165, −13.61743738309376570751955126605, −12.26056275226374829813478568103, −11.93210688403873379200367484355, −10.33463719896245463864248019050, −9.102364460192923616922099823585, −7.42654308351057545351148936149, −5.47269875881159923417992592015, −4.39078040069180434001235304876, −2.11602028355029869269824802902, 3.41765278402551306936834690468, 4.77380015332121113079951627505, 6.78065580216907961166213731846, 7.38888425082578096367451793073, 9.146059871424469212622863784358, 10.86673149431387620859154719467, 11.94781794366788052850281006088, 13.13535783305270661243027843406, 14.44130411113307396722600339647, 14.88183617865219314441327156447

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