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Properties

Label 2-448-112.109-c1-0-5
Degree $2$
Conductor $448$
Sign $0.757 - 0.652i$
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  + (−0.827 + 0.221i)3-s + (4.10 + 1.10i)5-s + (2.50 + 0.856i)7-s + (−1.96 + 1.13i)9-s + (0.318 + 1.18i)11-s + (−1.73 − 1.73i)13-s − 3.64·15-s + (−0.931 + 1.61i)17-s + (0.989 − 3.69i)19-s + (−2.26 − 0.153i)21-s + (−1.23 + 0.711i)23-s + (11.3 + 6.54i)25-s + (3.19 − 3.19i)27-s + (−0.181 − 0.181i)29-s + (−3.23 + 5.59i)31-s + ⋯
L(s)  = 1  + (−0.477 + 0.128i)3-s + (1.83 + 0.492i)5-s + (0.946 + 0.323i)7-s + (−0.653 + 0.377i)9-s + (0.0960 + 0.358i)11-s + (−0.480 − 0.480i)13-s − 0.941·15-s + (−0.226 + 0.391i)17-s + (0.227 − 0.847i)19-s + (−0.493 − 0.0335i)21-s + (−0.257 + 0.148i)23-s + (2.26 + 1.30i)25-s + (0.614 − 0.614i)27-s + (−0.0337 − 0.0337i)29-s + (−0.580 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.757 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54768 + 0.574344i\)
\(L(\frac12)\) \(\approx\) \(1.54768 + 0.574344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 - 0.856i)T \)
good3 \( 1 + (0.827 - 0.221i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-4.10 - 1.10i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.318 - 1.18i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.73 + 1.73i)T + 13iT^{2} \)
17 \( 1 + (0.931 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.989 + 3.69i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.23 - 0.711i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.181 + 0.181i)T + 29iT^{2} \)
31 \( 1 + (3.23 - 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.237 - 0.0637i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.440iT - 41T^{2} \)
43 \( 1 + (-5.54 + 5.54i)T - 43iT^{2} \)
47 \( 1 + (-3.61 - 6.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.59 + 9.69i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.438 - 1.63i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.41 + 9.01i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (9.59 - 2.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.2iT - 71T^{2} \)
73 \( 1 + (8.13 + 4.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.52 + 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.06 - 3.06i)T + 83iT^{2} \)
89 \( 1 + (5.66 - 3.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.70T + 97T^{2} \)
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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

−10.90009995179455699982657924336, −10.51623963258826180412745148133, −9.470295433361858299430722866577, −8.686792196496082447302371673751, −7.40515381841346558840970598094, −6.28513246589633255940238374434, −5.47656326296785372741075716551, −4.87349061654308602005873717878, −2.76630990840418350067551816353, −1.81845310948882258206130854516, 1.27265962104585497553803838229, 2.48729778270327140945672030714, 4.42418135534290860262758290471, 5.55400868443492424816137317223, 5.95387960049505409427435288521, 7.15409352699469715214646950432, 8.480883200017449100642190006486, 9.251445842375019799111203004399, 10.06307326965922080111567287355, 10.99859687406745710434534735961

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