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Properties

Label 2-448-7.5-c2-0-14
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: $\chi_{448} (257, \cdot )$
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

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

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