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Properties

Label 2-448-112.69-c2-0-1
Degree $2$
Conductor $448$
Sign $0.501 - 0.865i$
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  + (−1.75 − 1.75i)3-s + (−3.25 + 3.25i)5-s + (−4.90 − 4.99i)7-s − 2.83i·9-s + (2.01 + 2.01i)11-s + (−6.07 − 6.07i)13-s + 11.4·15-s + 9.55i·17-s + (25.2 + 25.2i)19-s + (−0.155 + 17.3i)21-s − 14.0i·23-s + 3.79i·25-s + (−20.7 + 20.7i)27-s + (−11.2 + 11.2i)29-s + 16.5i·31-s + ⋯
L(s)  = 1  + (−0.585 − 0.585i)3-s + (−0.651 + 0.651i)5-s + (−0.700 − 0.713i)7-s − 0.315i·9-s + (0.182 + 0.182i)11-s + (−0.467 − 0.467i)13-s + 0.762·15-s + 0.562i·17-s + (1.32 + 1.32i)19-s + (−0.00738 + 0.827i)21-s − 0.610i·23-s + 0.151i·25-s + (−0.769 + 0.769i)27-s + (−0.386 + 0.386i)29-s + 0.532i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.501 - 0.865i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.501 - 0.865i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7226480715\)
\(L(\frac12)\) \(\approx\) \(0.7226480715\)
\(L(2)\) 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 \( 1 + (4.90 + 4.99i)T \)
good3 \( 1 + (1.75 + 1.75i)T + 9iT^{2} \)
5 \( 1 + (3.25 - 3.25i)T - 25iT^{2} \)
11 \( 1 + (-2.01 - 2.01i)T + 121iT^{2} \)
13 \( 1 + (6.07 + 6.07i)T + 169iT^{2} \)
17 \( 1 - 9.55iT - 289T^{2} \)
19 \( 1 + (-25.2 - 25.2i)T + 361iT^{2} \)
23 \( 1 + 14.0iT - 529T^{2} \)
29 \( 1 + (11.2 - 11.2i)T - 841iT^{2} \)
31 \( 1 - 16.5iT - 961T^{2} \)
37 \( 1 + (-27.6 - 27.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 12.9T + 1.68e3T^{2} \)
43 \( 1 + (-50.9 - 50.9i)T + 1.84e3iT^{2} \)
47 \( 1 + 66.5iT - 2.20e3T^{2} \)
53 \( 1 + (-39.3 - 39.3i)T + 2.80e3iT^{2} \)
59 \( 1 + (30.4 - 30.4i)T - 3.48e3iT^{2} \)
61 \( 1 + (11.9 + 11.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-19.5 + 19.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 69.7iT - 5.04e3T^{2} \)
73 \( 1 - 143.T + 5.32e3T^{2} \)
79 \( 1 + 66.7T + 6.24e3T^{2} \)
83 \( 1 + (60.7 + 60.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 78.9T + 7.92e3T^{2} \)
97 \( 1 + 35.4iT - 9.40e3T^{2} \)
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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

−11.07784895608252829878490502114, −10.25137983899579532053682475406, −9.435455355121750810496696964619, −7.947721057099404465082093014635, −7.24399597181335080009893243441, −6.53166254218427721316631330677, −5.55800319689093621632290309208, −3.97269350435263952929492790507, −3.12722597948199561129456977424, −1.10356032429979591792114421227, 0.39056016551783048795622725958, 2.58024846202079488067958570461, 4.01415929016148350767381071838, 4.99728380746442858283445607640, 5.72947560643828579550044424287, 7.04604964673186980156035568244, 8.004966183188263605373478996023, 9.326824686821132418227528358441, 9.515487402514060205007165542596, 10.93269712109442337040684368760

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