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Properties

Label 2-448-7.2-c1-0-2
Degree $2$
Conductor $448$
Sign $0.968 - 0.250i$
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−2 + 1.73i)7-s + (1 − 1.73i)9-s + (1.5 + 2.59i)11-s + 6·13-s + 0.999·15-s + (2.5 + 4.33i)17-s + (0.5 − 0.866i)19-s + (2.5 + 0.866i)21-s + (3.5 − 6.06i)23-s + (2 + 3.46i)25-s − 5·27-s − 2·29-s + (2.5 + 4.33i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.755 + 0.654i)7-s + (0.333 − 0.577i)9-s + (0.452 + 0.783i)11-s + 1.66·13-s + 0.258·15-s + (0.606 + 1.05i)17-s + (0.114 − 0.198i)19-s + (0.545 + 0.188i)21-s + (0.729 − 1.26i)23-s + (0.400 + 0.692i)25-s − 0.962·27-s − 0.371·29-s + (0.449 + 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24894 + 0.159157i\)
\(L(\frac12)\) \(\approx\) \(1.24894 + 0.159157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (2.5 - 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.5 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.17472346800257854414717563573, −10.32182053853069587911626101136, −9.248502463746238962950567047845, −8.524708730523330474502969610910, −7.18325214464915713201215110724, −6.48461306680478037712395659960, −5.78480789819972127917308638266, −4.12021583864442773724023589305, −3.11156906286050994004484301992, −1.36245434682377856455429550156, 1.02716428366983755191768007058, 3.29588785858334462259153216482, 4.10686815133672853469954152287, 5.32006504234613265303786687791, 6.28460647726241375220285160019, 7.39144969507486566533369515328, 8.397903179338401172075113885829, 9.394529252350323576506991990101, 10.15811908975369985644136033384, 11.13263748987796882547345526943

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