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Properties

Label 2-448-7.4-c3-0-26
Degree $2$
Conductor $448$
Sign $0.968 + 0.250i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.5 − 6.06i)3-s + (3.5 + 6.06i)5-s + (14 + 12.1i)7-s + (−11 − 19.0i)9-s + (−2.5 + 4.33i)11-s + 14·13-s + 49·15-s + (10.5 − 18.1i)17-s + (24.5 + 42.4i)19-s + (122.5 − 42.4i)21-s + (79.5 + 137. i)23-s + (38 − 65.8i)25-s + 35.0·27-s − 58·29-s + (−73.5 + 127. i)31-s + ⋯
L(s)  = 1  + (0.673 − 1.16i)3-s + (0.313 + 0.542i)5-s + (0.755 + 0.654i)7-s + (−0.407 − 0.705i)9-s + (−0.0685 + 0.118i)11-s + 0.298·13-s + 0.843·15-s + (0.149 − 0.259i)17-s + (0.295 + 0.512i)19-s + (1.27 − 0.440i)21-s + (0.720 + 1.24i)23-s + (0.303 − 0.526i)25-s + 0.249·27-s − 0.371·29-s + (−0.425 + 0.737i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.958571531\)
\(L(\frac12)\) \(\approx\) \(2.958571531\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14 - 12.1i)T \)
good3 \( 1 + (-3.5 + 6.06i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-3.5 - 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 14T + 2.19e3T^{2} \)
17 \( 1 + (-10.5 + 18.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-24.5 - 42.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-79.5 - 137. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 58T + 2.43e4T^{2} \)
31 \( 1 + (73.5 - 127. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-109.5 - 189. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 350T + 6.89e4T^{2} \)
43 \( 1 - 124T + 7.95e4T^{2} \)
47 \( 1 + (262.5 + 454. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-151.5 + 262. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (52.5 - 90.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (206.5 + 357. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-207.5 + 359. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 432T + 3.57e5T^{2} \)
73 \( 1 + (-556.5 + 963. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-51.5 - 89.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + (-164.5 - 284. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 882T + 9.12e5T^{2} \)
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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

−10.76747617433277539499767509063, −9.577533983811628467045741351173, −8.640023888429254565598434196152, −7.83857085092552229642783276265, −7.11682628848754848102137056665, −6.10234684082523710500975739330, −5.02758536497054837990160369981, −3.29347214584477096785251715431, −2.25714262697305759650000518311, −1.31301168857112923451803431646, 1.05546947524937918372165883782, 2.71170561795080450864406798548, 4.02502302225733547144911770211, 4.65368469215847972903208565599, 5.71253535938362590178342610061, 7.20770930241592844680741901886, 8.247768229549857085320458497534, 9.040020581717548970865301990785, 9.661974387392029638717485883377, 10.75452950185204411768887184738

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