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Properties

Label 2-56-56.45-c2-0-2
Degree $2$
Conductor $56$
Sign $0.443 - 0.896i$
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  + (−1.87 + 0.705i)2-s + (−0.126 + 0.219i)3-s + (3.00 − 2.64i)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 + (−5.52 − 4.52i)10-s + (−6.82 − 3.94i)11-s + (0.198 + 0.993i)12-s + 18.1·13-s + (−9.91 − 9.88i)14-s − 0.904·15-s + (2.04 − 15.8i)16-s + (−8.26 − 4.76i)17-s + ⋯
L(s)  = 1  + (−0.935 + 0.352i)2-s + (−0.0422 + 0.0731i)3-s + (0.750 − 0.660i)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 + (−0.552 − 0.452i)10-s + (−0.620 − 0.358i)11-s + (0.0165 + 0.0827i)12-s + 1.39·13-s + (−0.708 − 0.706i)14-s − 0.0603·15-s + (0.127 − 0.991i)16-s + (−0.485 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.443 - 0.896i$
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: \((2,\ 56,\ (\ :1),\ 0.443 - 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.721232 + 0.447846i\)
\(L(\frac12)\) \(\approx\) \(0.721232 + 0.447846i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.87 - 0.705i)T \)
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

−15.57357888334082354075673208231, −14.35891737378464331230302663472, −13.08220261688042084878354961513, −11.14372130759514995432409084899, −10.73803593253937803752680357287, −9.164565038634921982268645728463, −8.144157754743887345592062229641, −6.68898667507114350426593776801, −5.36691169824807992282219962817, −2.30548874959915775492077229833, 1.34678431146339139567362300603, 3.98134897150763780742369869127, 6.32479539144710595928419170411, 7.77435021330422406907066740441, 8.928767181740523652420039895756, 10.15346679112694005229311433219, 11.10085483341901048366969590579, 12.54020226370235189892698670843, 13.35060361657366957891090886791, 15.08115614456246828521996031698

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