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Properties

Label 2-448-16.13-c1-0-2
Degree $2$
Conductor $448$
Sign $0.992 - 0.118i$
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  + (−1.20 − 1.20i)3-s + (−2.59 + 2.59i)5-s − i·7-s − 0.0802i·9-s + (1.08 − 1.08i)11-s + (2.97 + 2.97i)13-s + 6.26·15-s + 7.18·17-s + (2.25 + 2.25i)19-s + (−1.20 + 1.20i)21-s + 3.49i·23-s − 8.45i·25-s + (−3.72 + 3.72i)27-s + (0.851 + 0.851i)29-s + 3.95·31-s + ⋯
L(s)  = 1  + (−0.697 − 0.697i)3-s + (−1.15 + 1.15i)5-s − 0.377i·7-s − 0.0267i·9-s + (0.325 − 0.325i)11-s + (0.826 + 0.826i)13-s + 1.61·15-s + 1.74·17-s + (0.517 + 0.517i)19-s + (−0.263 + 0.263i)21-s + 0.729i·23-s − 1.69i·25-s + (−0.716 + 0.716i)27-s + (0.158 + 0.158i)29-s + 0.710·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.992 - 0.118i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.992 - 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942372 + 0.0562469i\)
\(L(\frac12)\) \(\approx\) \(0.942372 + 0.0562469i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + (1.20 + 1.20i)T + 3iT^{2} \)
5 \( 1 + (2.59 - 2.59i)T - 5iT^{2} \)
11 \( 1 + (-1.08 + 1.08i)T - 11iT^{2} \)
13 \( 1 + (-2.97 - 2.97i)T + 13iT^{2} \)
17 \( 1 - 7.18T + 17T^{2} \)
19 \( 1 + (-2.25 - 2.25i)T + 19iT^{2} \)
23 \( 1 - 3.49iT - 23T^{2} \)
29 \( 1 + (-0.851 - 0.851i)T + 29iT^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
37 \( 1 + (-5.93 + 5.93i)T - 37iT^{2} \)
41 \( 1 - 2.67iT - 41T^{2} \)
43 \( 1 + (-4.25 + 4.25i)T - 43iT^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 + (3.41 - 3.41i)T - 53iT^{2} \)
59 \( 1 + (3.86 - 3.86i)T - 59iT^{2} \)
61 \( 1 + (-1.40 - 1.40i)T + 61iT^{2} \)
67 \( 1 + (5.70 + 5.70i)T + 67iT^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + 3.32iT - 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + (1.20 + 1.20i)T + 83iT^{2} \)
89 \( 1 - 12.6iT - 89T^{2} \)
97 \( 1 - 1.08T + 97T^{2} \)
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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

−11.35396630183371142208784285284, −10.52644210301319038307082886780, −9.399512462272706591207945858247, −7.962207584900124078063002500769, −7.39030019406129046079114525719, −6.54010662326275933992363813459, −5.75054531206602067070343213503, −3.98290742331861530890238303769, −3.26671357230203089937359661728, −1.11046373106550173185892475258, 0.887755192911159114955102263313, 3.32046696490336891088959448666, 4.47345090414487886391998067064, 5.13105757485188291793319690907, 6.09699415681366837937529276086, 7.75438804727329224467014405156, 8.231253926008790286388756829035, 9.341724775373964766850672019195, 10.23094152346223692094402704629, 11.21812317465128505658794294902

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