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Properties

Label 2-448-112.83-c1-0-5
Degree $2$
Conductor $448$
Sign $0.437 - 0.899i$
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  + (2.23 + 2.23i)3-s + (−0.584 − 0.584i)5-s + (1.82 − 1.91i)7-s + 6.98i·9-s + (2 + 2i)11-s + (−2.91 + 2.91i)13-s − 2.61i·15-s − 2.66i·17-s + (0.319 + 0.319i)19-s + (8.35 − 0.198i)21-s + 3.27·23-s − 4.31i·25-s + (−8.90 + 8.90i)27-s + (−2.04 − 2.04i)29-s + 2.52·31-s + ⋯
L(s)  = 1  + (1.29 + 1.29i)3-s + (−0.261 − 0.261i)5-s + (0.690 − 0.723i)7-s + 2.32i·9-s + (0.603 + 0.603i)11-s + (−0.807 + 0.807i)13-s − 0.673i·15-s − 0.645i·17-s + (0.0733 + 0.0733i)19-s + (1.82 − 0.0432i)21-s + 0.683·23-s − 0.863i·25-s + (−1.71 + 1.71i)27-s + (−0.379 − 0.379i)29-s + 0.453·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79724 + 1.12450i\)
\(L(\frac12)\) \(\approx\) \(1.79724 + 1.12450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 + 1.91i)T \)
good3 \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \)
5 \( 1 + (0.584 + 0.584i)T + 5iT^{2} \)
11 \( 1 + (-2 - 2i)T + 11iT^{2} \)
13 \( 1 + (2.91 - 2.91i)T - 13iT^{2} \)
17 \( 1 + 2.66iT - 17T^{2} \)
19 \( 1 + (-0.319 - 0.319i)T + 19iT^{2} \)
23 \( 1 - 3.27T + 23T^{2} \)
29 \( 1 + (2.04 + 2.04i)T + 29iT^{2} \)
31 \( 1 - 2.52T + 31T^{2} \)
37 \( 1 + (-1.70 + 1.70i)T - 37iT^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + (-3.27 - 3.27i)T + 43iT^{2} \)
47 \( 1 + 9.96T + 47T^{2} \)
53 \( 1 + (2.37 - 2.37i)T - 53iT^{2} \)
59 \( 1 + (-4.14 + 4.14i)T - 59iT^{2} \)
61 \( 1 + (-4.52 + 4.52i)T - 61iT^{2} \)
67 \( 1 + (-4.37 + 4.37i)T - 67iT^{2} \)
71 \( 1 + 5.14T + 71T^{2} \)
73 \( 1 - 6.99T + 73T^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + (5.39 + 5.39i)T + 83iT^{2} \)
89 \( 1 - 1.05T + 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−11.11513277020995475941379625037, −10.01833811716582092765657772921, −9.567225883765016343397632216842, −8.641019840525064833580782506842, −7.84665663657424514343726320801, −6.92699452601602996290212905687, −4.82355650349521553041677347775, −4.52313677231321831034213006390, −3.44190206202446917152798215338, −2.05407217009127954767493783362, 1.42763934872208837966946495510, 2.65331632871957505146499618708, 3.56226793265106854708498451183, 5.32325284461433580963299175267, 6.57409470532584200318018382272, 7.40460091241359930887349678069, 8.270529415046391625745533558549, 8.730491174538219674612010014113, 9.776378213766847349336257594206, 11.20252392252458071712901966561

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