Properties

Label 2-56-56.53-c3-0-14
Degree $2$
Conductor $56$
Sign $0.936 + 0.350i$
Analytic cond. $3.30410$
Root an. cond. $1.81772$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.53 + 1.25i)2-s + (−7.07 − 4.08i)3-s + (4.85 + 6.35i)4-s + (17.0 − 9.82i)5-s + (−12.8 − 19.2i)6-s + (15.6 − 9.83i)7-s + (4.34 + 22.2i)8-s + (19.9 + 34.4i)9-s + (55.4 − 3.57i)10-s + (−29.7 − 17.1i)11-s + (−8.39 − 64.8i)12-s − 3.09i·13-s + (52.1 − 5.26i)14-s − 160.·15-s + (−16.8 + 61.7i)16-s + (−14.8 + 25.6i)17-s + ⋯
L(s)  = 1  + (0.896 + 0.443i)2-s + (−1.36 − 0.786i)3-s + (0.607 + 0.794i)4-s + (1.52 − 0.879i)5-s + (−0.872 − 1.30i)6-s + (0.847 − 0.531i)7-s + (0.191 + 0.981i)8-s + (0.737 + 1.27i)9-s + (1.75 − 0.113i)10-s + (−0.814 − 0.470i)11-s + (−0.202 − 1.56i)12-s − 0.0661i·13-s + (0.994 − 0.100i)14-s − 2.76·15-s + (−0.262 + 0.964i)16-s + (−0.211 + 0.365i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(3.30410\)
Root analytic conductor: \(1.81772\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :3/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.88937 - 0.341468i\)
\(L(\frac12)\) \(\approx\) \(1.88937 - 0.341468i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.53 - 1.25i)T \)
7 \( 1 + (-15.6 + 9.83i)T \)
good3 \( 1 + (7.07 + 4.08i)T + (13.5 + 23.3i)T^{2} \)
5 \( 1 + (-17.0 + 9.82i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (29.7 + 17.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 3.09iT - 2.19e3T^{2} \)
17 \( 1 + (14.8 - 25.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (40.0 - 23.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-49.7 - 86.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 136. iT - 2.43e4T^{2} \)
31 \( 1 + (0.290 - 0.502i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (128. - 74.0i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 141.T + 6.89e4T^{2} \)
43 \( 1 - 259. iT - 7.95e4T^{2} \)
47 \( 1 + (174. + 301. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (184. + 106. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (100. + 57.9i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-345. + 199. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-893. - 515. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 476.T + 3.57e5T^{2} \)
73 \( 1 + (554. - 960. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (225. + 391. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 770. iT - 5.71e5T^{2} \)
89 \( 1 + (615. + 1.06e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 706.T + 9.12e5T^{2} \)
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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

−14.33823025226148858139288701689, −13.22992477061628049643505786560, −12.86473471696076590446911418042, −11.52296595998513668940167547032, −10.47773950860523813929262868232, −8.311057036604191730516753763408, −6.78949071914508261647494819264, −5.61320224826642492280642668386, −5.00425526259837835091751741928, −1.62597726957999671798427521303, 2.31725461991407600992767223249, 4.80466372186571456559953868753, 5.61650252154704716553628556899, 6.65595958917547814699092960873, 9.650635631792153709694176667084, 10.58746922861339757268433832820, 11.15319685403056850462826234191, 12.43397460743115742269124528282, 13.71560686418139426657630261058, 14.81877921670701251769938842682

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