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Properties

Label 2-28-28.19-c3-0-1
Degree $2$
Conductor $28$
Sign $0.927 - 0.373i$
Analytic cond. $1.65205$
Root an. cond. $1.28532$
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  + (−2.82 + 0.117i)2-s + (0.0469 + 0.0812i)3-s + (7.97 − 0.662i)4-s + (12.4 + 7.20i)5-s + (−0.142 − 0.224i)6-s + (15.0 + 10.7i)7-s + (−22.4 + 2.80i)8-s + (13.4 − 23.3i)9-s + (−36.1 − 18.9i)10-s + (−31.6 + 18.2i)11-s + (0.428 + 0.616i)12-s − 15.3i·13-s + (−43.8 − 28.6i)14-s + 1.35i·15-s + (63.1 − 10.5i)16-s + (−54.1 + 31.2i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0414i)2-s + (0.00903 + 0.0156i)3-s + (0.996 − 0.0828i)4-s + (1.11 + 0.644i)5-s + (−0.00967 − 0.0152i)6-s + (0.813 + 0.581i)7-s + (−0.992 + 0.124i)8-s + (0.499 − 0.865i)9-s + (−1.14 − 0.597i)10-s + (−0.866 + 0.500i)11-s + (0.0102 + 0.0148i)12-s − 0.326i·13-s + (−0.837 − 0.547i)14-s + 0.0232i·15-s + (0.986 − 0.165i)16-s + (−0.772 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.931214 + 0.180510i\)
\(L(\frac12)\) \(\approx\) \(0.931214 + 0.180510i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 - 0.117i)T \)
7 \( 1 + (-15.0 - 10.7i)T \)
good3 \( 1 + (-0.0469 - 0.0812i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-12.4 - 7.20i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (31.6 - 18.2i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 15.3iT - 2.19e3T^{2} \)
17 \( 1 + (54.1 - 31.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-34.6 + 60.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-12.3 - 7.14i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 157.T + 2.43e4T^{2} \)
31 \( 1 + (165. + 286. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (38.0 - 65.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 335. iT - 6.89e4T^{2} \)
43 \( 1 - 484. iT - 7.95e4T^{2} \)
47 \( 1 + (-187. + 324. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-70.8 - 122. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-159. - 277. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-213. - 123. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-178. + 102. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 82.4iT - 3.57e5T^{2} \)
73 \( 1 + (575. - 331. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (564. + 325. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 790.T + 5.71e5T^{2} \)
89 \( 1 + (-631. - 364. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 386. iT - 9.12e5T^{2} \)
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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

−17.32526704183793172212494357804, −15.49128636544183498823080600148, −14.80905742437368977228522090240, −12.98675493558003324606125838103, −11.34840876440359965240918367195, −10.14678334520355735536937078210, −9.058182813221802116919379708363, −7.33928827935220038516778249633, −5.82369744959570489866634617049, −2.21682929291987103231494259271, 1.76656138600099395787018752922, 5.33598680193499329068811678356, 7.37987334620868244434481774877, 8.727600535076395669230725865542, 10.12305339366038472520412660227, 11.11303332921122059610760157509, 12.96459614723619615834971736864, 14.11115382360288255091564923329, 15.96992466593846281330838656599, 16.80953797435221575931349470020

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