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Properties

Label 2-448-56.3-c1-0-8
Degree $2$
Conductor $448$
Sign $0.572 - 0.819i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.43 + 0.827i)3-s + (2.02 + 3.50i)5-s + (2.64 − 0.129i)7-s + (−0.129 − 0.224i)9-s + (1.20 − 2.09i)11-s − 2.25·13-s + 6.70i·15-s + (−6.07 − 3.50i)17-s + (2.34 − 1.35i)19-s + (3.89 + 2.00i)21-s + (−4.30 + 2.48i)23-s + (−5.70 + 9.88i)25-s − 5.39i·27-s + 1.82i·29-s + (2.94 − 5.09i)31-s + ⋯
L(s)  = 1  + (0.827 + 0.477i)3-s + (0.905 + 1.56i)5-s + (0.998 − 0.0490i)7-s + (−0.0432 − 0.0749i)9-s + (0.364 − 0.631i)11-s − 0.626·13-s + 1.73i·15-s + (−1.47 − 0.850i)17-s + (0.537 − 0.310i)19-s + (0.850 + 0.436i)21-s + (−0.896 + 0.517i)23-s + (−1.14 + 1.97i)25-s − 1.03i·27-s + 0.338i·29-s + (0.528 − 0.914i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.572 - 0.819i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.572 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90953 + 0.995593i\)
\(L(\frac12)\) \(\approx\) \(1.90953 + 0.995593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.129i)T \)
good3 \( 1 + (-1.43 - 0.827i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.02 - 3.50i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.20 + 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.25T + 13T^{2} \)
17 \( 1 + (6.07 + 3.50i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.34 + 1.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.30 - 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.82iT - 29T^{2} \)
31 \( 1 + (-2.94 + 5.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 - 6.92T + 43T^{2} \)
47 \( 1 + (3.39 + 5.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.46 + 2.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.47 + 1.43i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.75 - 6.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.58 - 9.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (-2.68 - 1.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.85 - 2.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.89iT - 83T^{2} \)
89 \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
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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

−11.26169003506903027419933920603, −10.19910463995742894060113362114, −9.520827945805371010578167299939, −8.679227674515629358529424294163, −7.53435409418695002941213424444, −6.65799098846148388191396989195, −5.61981417307701426879719001672, −4.22842231337572559480130417917, −2.98479979816467626492039642552, −2.19969159877326189222750438842, 1.58591571522157125877283974505, 2.23257874515560823088151629882, 4.36499801445260648324769185197, 5.02645164925593839899699270124, 6.20131469129694279175549391076, 7.59716808486648854368039366926, 8.390752448859140680986146227666, 8.938451899141740185361275721336, 9.788151974168586061728185943087, 10.92943075640635325137879490283

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