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Properties

Label 2-448-112.109-c1-0-11
Degree $2$
Conductor $448$
Sign $0.916 + 0.400i$
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  + (2.46 − 0.659i)3-s + (1.87 + 0.502i)5-s + (−0.364 − 2.62i)7-s + (3.02 − 1.74i)9-s + (1.37 + 5.11i)11-s + (−3.61 − 3.61i)13-s + 4.94·15-s + (−0.294 + 0.510i)17-s + (0.137 − 0.514i)19-s + (−2.62 − 6.20i)21-s + (−0.378 + 0.218i)23-s + (−1.06 − 0.615i)25-s + (0.878 − 0.878i)27-s + (2.68 + 2.68i)29-s + (−3.94 + 6.82i)31-s + ⋯
L(s)  = 1  + (1.42 − 0.380i)3-s + (0.838 + 0.224i)5-s + (−0.137 − 0.990i)7-s + (1.00 − 0.581i)9-s + (0.413 + 1.54i)11-s + (−1.00 − 1.00i)13-s + 1.27·15-s + (−0.0715 + 0.123i)17-s + (0.0316 − 0.118i)19-s + (−0.572 − 1.35i)21-s + (−0.0788 + 0.0455i)23-s + (−0.213 − 0.123i)25-s + (0.169 − 0.169i)27-s + (0.499 + 0.499i)29-s + (−0.707 + 1.22i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.916 + 0.400i$
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: \((2,\ 448,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31495 - 0.484432i\)
\(L(\frac12)\) \(\approx\) \(2.31495 - 0.484432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.364 + 2.62i)T \)
good3 \( 1 + (-2.46 + 0.659i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.87 - 0.502i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.37 - 5.11i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.61 + 3.61i)T + 13iT^{2} \)
17 \( 1 + (0.294 - 0.510i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.137 + 0.514i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.378 - 0.218i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 - 2.68i)T + 29iT^{2} \)
31 \( 1 + (3.94 - 6.82i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.90 + 0.779i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.95iT - 41T^{2} \)
43 \( 1 + (-7.67 + 7.67i)T - 43iT^{2} \)
47 \( 1 + (2.01 + 3.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.244 - 0.912i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.00753 + 0.0281i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.529 + 1.97i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-6.24 + 1.67i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.05iT - 71T^{2} \)
73 \( 1 + (-6.85 - 3.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.53 - 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (3.35 - 1.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−10.65291116473518427872855561245, −9.930470302945682362798982719615, −9.396619499873170246448578272793, −8.244501695508188537293975650027, −7.28976878778509116081398784699, −6.85753830607458846984368522990, −5.20127012134626036608445176103, −3.93007318044246843207314511541, −2.72771793074420037731983041606, −1.72159307724073745944440877847, 2.03911900522861504222223472722, 2.90033283453615254917823474618, 4.10745504478766663589053593023, 5.48880732605527093130828275340, 6.38095491229262339339062679763, 7.82548795044676487501949072838, 8.689110957406474519746017282780, 9.372999829385604381149349695236, 9.692760259314485419417729072964, 11.14658411750634287916228281065

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