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Properties

Label 2-448-28.19-c3-0-38
Degree $2$
Conductor $448$
Sign $-0.0742 + 0.997i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
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  + (3.53 + 6.11i)3-s + (−1.05 − 0.611i)5-s + (−16.5 − 8.40i)7-s + (−11.4 + 19.8i)9-s + (3.96 − 2.28i)11-s − 41.0i·13-s − 8.64i·15-s + (−103. + 59.7i)17-s + (30.0 − 52.1i)19-s + (−6.85 − 130. i)21-s + (7.23 + 4.17i)23-s + (−61.7 − 106. i)25-s + 29.0·27-s − 164.·29-s + (−143. − 249. i)31-s + ⋯
L(s)  = 1  + (0.679 + 1.17i)3-s + (−0.0948 − 0.0547i)5-s + (−0.891 − 0.453i)7-s + (−0.423 + 0.733i)9-s + (0.108 − 0.0627i)11-s − 0.876i·13-s − 0.148i·15-s + (−1.47 + 0.852i)17-s + (0.363 − 0.629i)19-s + (−0.0712 − 1.35i)21-s + (0.0655 + 0.0378i)23-s + (−0.494 − 0.855i)25-s + 0.207·27-s − 1.05·29-s + (−0.833 − 1.44i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0742 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0742 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.0742 + 0.997i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.0742 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7561943777\)
\(L(\frac12)\) \(\approx\) \(0.7561943777\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (16.5 + 8.40i)T \)
good3 \( 1 + (-3.53 - 6.11i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (1.05 + 0.611i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (-3.96 + 2.28i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 41.0iT - 2.19e3T^{2} \)
17 \( 1 + (103. - 59.7i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-30.0 + 52.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-7.23 - 4.17i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 164.T + 2.43e4T^{2} \)
31 \( 1 + (143. + 249. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (74.3 - 128. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 358. iT - 6.89e4T^{2} \)
43 \( 1 + 360. iT - 7.95e4T^{2} \)
47 \( 1 + (112. - 195. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-342. - 592. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-42.5 - 73.6i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-181. - 104. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-361. + 208. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 982. iT - 3.57e5T^{2} \)
73 \( 1 + (-295. + 170. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (66.0 + 38.1i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 523.T + 5.71e5T^{2} \)
89 \( 1 + (841. + 486. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 676. iT - 9.12e5T^{2} \)
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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

−10.44627625332303516099654026713, −9.444421233071459648921558774848, −8.949631097600358495565887672652, −7.86020431360945580344448052964, −6.72237525537695755335943307740, −5.57172862101864198139369656558, −4.21673637566036278828424820041, −3.66843429162971075196147983656, −2.45983541118618373788721977817, −0.20900474593128644689055307226, 1.60970065482185188728057903331, 2.60977599736772476314857897331, 3.76955692542206893888037312599, 5.34403698766255430220178015540, 6.74483800589038136747913407004, 6.97112031027335393717601229790, 8.178058256852115950470566520011, 9.069698108774450196304854071673, 9.678398472839106453619892423669, 11.16620866638229792665515452309

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