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Properties

Label 2-448-7.4-c1-0-7
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: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67420 - 0.213349i\)
\(L(\frac12)\) \(\approx\) \(1.67420 - 0.213349i\)
\(L(\frac{3}{2})\) 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

−11.06570416477303644862882219526, −10.28112060515923662557039706769, −8.981284766262318275552774781135, −8.280377162677080167753195997936, −7.61173860775995476336679969347, −6.45734543381002655017344739196, −5.21121549882530331829276896099, −4.38258199261233761556865705868, −2.69041184750624885520527305361, −1.46536509261020015553381272530, 1.41074916071950815379376645270, 3.58364774056808909538455541099, 3.82341050875473584304683662256, 5.44704265117826501226717036366, 6.41014982737963124489172099919, 7.68384655327434480624255997020, 8.337485572470501758868547458266, 9.336573521484889522234571637343, 10.44546283974423156756024697187, 10.95012895598833698182073528060

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