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Properties

Label 2-448-7.2-c1-0-7
Degree $2$
Conductor $448$
Sign $0.922 - 0.386i$
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  + (0.866 + 1.5i)3-s + (0.5 − 0.866i)5-s + (1.73 − 2i)7-s + (2.59 + 4.5i)11-s + 1.73·15-s + (−2.5 − 4.33i)17-s + (0.866 − 1.5i)19-s + (4.5 + 0.866i)21-s + (0.866 − 1.5i)23-s + (2 + 3.46i)25-s + 5.19·27-s − 8·29-s + (4.33 + 7.5i)31-s + (−4.5 + 7.79i)33-s + (−0.866 − 2.5i)35-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.223 − 0.387i)5-s + (0.654 − 0.755i)7-s + (0.783 + 1.35i)11-s + 0.447·15-s + (−0.606 − 1.05i)17-s + (0.198 − 0.344i)19-s + (0.981 + 0.188i)21-s + (0.180 − 0.312i)23-s + (0.400 + 0.692i)25-s + 1.00·27-s − 1.48·29-s + (0.777 + 1.34i)31-s + (−0.783 + 1.35i)33-s + (−0.146 − 0.422i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.922 - 0.386i$
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: \((2,\ 448,\ (\ :1/2),\ 0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83451 + 0.368865i\)
\(L(\frac12)\) \(\approx\) \(1.83451 + 0.368865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 2i)T \)
good3 \( 1 + (-0.866 - 1.5i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.59 - 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-4.33 - 7.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 6.92T + 43T^{2} \)
47 \( 1 + (4.33 - 7.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (7.5 + 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.866 + 1.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−11.04199337693881045275987182948, −10.05371720488277029884326546150, −9.390995279942149073967908681464, −8.747599517131785321967761055489, −7.42905379687482957983230238156, −6.70854701317858839821165384533, −4.85983797560327033523500560990, −4.55303842784167273196598710644, −3.28332726545800663096597365968, −1.55436286452054426289403976863, 1.56004751391240996199052382506, 2.63240326008029503075519965577, 4.00968015610822571681707353223, 5.61128676605861189345793907248, 6.35483066505742571145785652903, 7.44619092933150430715551064251, 8.425433052368295310533097509366, 8.840865694908291369189647671996, 10.19997704203946212803611739263, 11.24352467458979583722636172253

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