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Properties

Label 2-448-7.3-c2-0-25
Degree $2$
Conductor $448$
Sign $0.314 + 0.949i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.438 + 0.253i)3-s + (4.59 − 2.65i)5-s + (5.27 − 4.59i)7-s + (−4.37 − 7.57i)9-s + (8.54 − 14.8i)11-s + 21.4i·13-s + 2.68·15-s + (−20.7 − 12.0i)17-s + (−10.5 + 6.11i)19-s + (3.48 − 0.680i)21-s + (−20.1 − 34.8i)23-s + (1.55 − 2.69i)25-s − 8.99i·27-s + 26.0·29-s + (21.8 + 12.6i)31-s + ⋯
L(s)  = 1  + (0.146 + 0.0844i)3-s + (0.918 − 0.530i)5-s + (0.753 − 0.656i)7-s + (−0.485 − 0.841i)9-s + (0.776 − 1.34i)11-s + 1.65i·13-s + 0.179·15-s + (−1.22 − 0.706i)17-s + (−0.557 + 0.321i)19-s + (0.165 − 0.0324i)21-s + (−0.875 − 1.51i)23-s + (0.0623 − 0.107i)25-s − 0.333i·27-s + 0.899·29-s + (0.705 + 0.407i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.314 + 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.116637627\)
\(L(\frac12)\) \(\approx\) \(2.116637627\)
\(L(2)\) 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 \( 1 + (-5.27 + 4.59i)T \)
good3 \( 1 + (-0.438 - 0.253i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-4.59 + 2.65i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.54 + 14.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 21.4iT - 169T^{2} \)
17 \( 1 + (20.7 + 12.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10.5 - 6.11i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (20.1 + 34.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 26.0T + 841T^{2} \)
31 \( 1 + (-21.8 - 12.6i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-6.48 - 11.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 33.8iT - 1.68e3T^{2} \)
43 \( 1 - 29.9T + 1.84e3T^{2} \)
47 \( 1 + (-48.2 + 27.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (4.36 - 7.55i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (43.2 + 24.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (3.40 - 1.96i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-52.9 + 91.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 35.0T + 5.04e3T^{2} \)
73 \( 1 + (-40.3 - 23.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-43.4 - 75.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 64.0iT - 6.88e3T^{2} \)
89 \( 1 + (-37.2 + 21.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 28.7iT - 9.40e3T^{2} \)
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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

−10.81077593410956445178465442693, −9.600073575912968689178516012171, −8.876368381988148476629548917946, −8.351296438987260549241148081182, −6.58923526383843744885834709069, −6.27548107952291330239922737946, −4.75755237176073977838455904080, −3.94444992802499522474703110852, −2.25913848737850604354203662146, −0.889570438394938266482534593529, 1.88324665052481397424507576436, 2.56484640688780431671805814692, 4.35284279150752646605019920950, 5.47817787872069338444265838809, 6.23496096357798322734775683108, 7.49299553654743541816364910095, 8.311850588500197202525356175157, 9.286635779192060342213361787301, 10.27339538137466917129810828727, 10.88971837545128144310270206556

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