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Properties

Label 2-56-8.3-c2-0-10
Degree $2$
Conductor $56$
Sign $0.467 + 0.883i$
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  + (0.707 − 1.87i)2-s + 3.41·3-s + (−3 − 2.64i)4-s + 1.54i·5-s + (2.41 − 6.38i)6-s + 2.64i·7-s + (−7.07 + 3.74i)8-s + 2.65·9-s + (2.89 + 1.09i)10-s − 4.48·11-s + (−10.2 − 9.03i)12-s − 1.54i·13-s + (4.94 + 1.87i)14-s + 5.29i·15-s + (1.99 + 15.8i)16-s + 23.6·17-s + ⋯
L(s)  = 1  + (0.353 − 0.935i)2-s + 1.13·3-s + (−0.750 − 0.661i)4-s + 0.309i·5-s + (0.402 − 1.06i)6-s + 0.377i·7-s + (−0.883 + 0.467i)8-s + 0.295·9-s + (0.289 + 0.109i)10-s − 0.407·11-s + (−0.853 − 0.752i)12-s − 0.119i·13-s + (0.353 + 0.133i)14-s + 0.352i·15-s + (0.124 + 0.992i)16-s + 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.467 + 0.883i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.467 + 0.883i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.41015 - 0.849227i\)
\(L(\frac12)\) \(\approx\) \(1.41015 - 0.849227i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.87i)T \)
7 \( 1 - 2.64iT \)
good3 \( 1 - 3.41T + 9T^{2} \)
5 \( 1 - 1.54iT - 25T^{2} \)
11 \( 1 + 4.48T + 121T^{2} \)
13 \( 1 + 1.54iT - 169T^{2} \)
17 \( 1 - 23.6T + 289T^{2} \)
19 \( 1 + 24.8T + 361T^{2} \)
23 \( 1 - 35.2iT - 529T^{2} \)
29 \( 1 + 22.4iT - 841T^{2} \)
31 \( 1 + 46.7iT - 961T^{2} \)
37 \( 1 + 58.5iT - 1.36e3T^{2} \)
41 \( 1 + 26.9T + 1.68e3T^{2} \)
43 \( 1 + 17.1T + 1.84e3T^{2} \)
47 \( 1 + 36.1iT - 2.20e3T^{2} \)
53 \( 1 - 97.8iT - 2.80e3T^{2} \)
59 \( 1 - 61.5T + 3.48e3T^{2} \)
61 \( 1 + 37.6iT - 3.72e3T^{2} \)
67 \( 1 + 33.3T + 4.48e3T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 - 69.3T + 5.32e3T^{2} \)
79 \( 1 - 38.7iT - 6.24e3T^{2} \)
83 \( 1 - 3.61T + 6.88e3T^{2} \)
89 \( 1 - 44.0T + 7.92e3T^{2} \)
97 \( 1 - 96.1T + 9.40e3T^{2} \)
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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

−14.66464782657398916987232444332, −13.67703513169460647213927227859, −12.71481669412572134793934677770, −11.43346997165337902442250985068, −10.13263843618171904115196657422, −9.073629248657825249060719661512, −7.88752687774977715756055899219, −5.65383144237285421528661073479, −3.68945261654229939025967584491, −2.38537397557832318135150549805, 3.24226836158359681892955663159, 4.87893051937379340550042834344, 6.70113376125101435710979255570, 8.129977058773498423303766773295, 8.751158734501348402137093035225, 10.25001156554853417951954896232, 12.36224363266777665519786194857, 13.27647926596480752812207193091, 14.40926651765945632170538833189, 14.83628199894067279647520742833

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