Properties

Label 2-28-28.3-c3-0-7
Degree $2$
Conductor $28$
Sign $0.301 + 0.953i$
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  + (1.31 − 2.50i)2-s + (−0.0469 + 0.0812i)3-s + (−4.56 − 6.57i)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 + (−1.69 − 40.7i)10-s + (31.6 + 18.2i)11-s + (0.748 − 0.0621i)12-s + 15.3i·13-s + (7.22 + 51.8i)14-s + 1.35i·15-s + (−22.4 + 59.9i)16-s + (−54.1 − 31.2i)17-s + ⋯
L(s)  = 1  + (0.463 − 0.886i)2-s + (−0.00903 + 0.0156i)3-s + (−0.570 − 0.821i)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 + (−0.0534 − 1.28i)10-s + (0.866 + 0.500i)11-s + (0.0180 − 0.00149i)12-s + 0.326i·13-s + (0.137 + 0.990i)14-s + 0.0232i·15-s + (−0.350 + 0.936i)16-s + (−0.772 − 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.301 + 0.953i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ 0.301 + 0.953i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20422 - 0.881980i\)
\(L(\frac12)\) \(\approx\) \(1.20422 - 0.881980i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 + 2.50i)T \)
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

−16.56636606066505257485143775608, −15.04958076376598776670888628257, −13.45835115822877280089313678552, −13.00880054062869549733425332062, −11.52939587483718343998571576102, −9.846814639992406271919361845794, −9.157450810184317133882339328307, −6.24864042056805407007202567573, −4.67505306978126347346995497015, −2.09715622940824625987543486802, 3.67584833602192963673360584173, 6.11005944561243325902106307348, 6.85147212231403325745140760337, 9.014652505318608512941780497396, 10.28204850126793859381287218075, 12.41546844120904551867741543076, 13.56100333459441254423671190270, 14.44474183166156132261286305121, 15.68662915373557808286560849593, 17.02775040546662327471014385593

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