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Properties

Label 2-448-112.69-c2-0-15
Degree $2$
Conductor $448$
Sign $0.237 + 0.971i$
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  + (1.11 + 1.11i)3-s + (−5.47 + 5.47i)5-s + (−3.11 − 6.26i)7-s − 6.53i·9-s + (−2.57 − 2.57i)11-s + (11.1 + 11.1i)13-s − 12.1·15-s − 24.6i·17-s + (−10.9 − 10.9i)19-s + (3.49 − 10.4i)21-s + 10.3i·23-s − 34.9i·25-s + (17.2 − 17.2i)27-s + (24.5 − 24.5i)29-s − 14.5i·31-s + ⋯
L(s)  = 1  + (0.370 + 0.370i)3-s + (−1.09 + 1.09i)5-s + (−0.445 − 0.895i)7-s − 0.725i·9-s + (−0.233 − 0.233i)11-s + (0.858 + 0.858i)13-s − 0.811·15-s − 1.44i·17-s + (−0.578 − 0.578i)19-s + (0.166 − 0.496i)21-s + 0.451i·23-s − 1.39i·25-s + (0.639 − 0.639i)27-s + (0.846 − 0.846i)29-s − 0.468i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.237 + 0.971i$
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: \((2,\ 448,\ (\ :1),\ 0.237 + 0.971i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9953493326\)
\(L(\frac12)\) \(\approx\) \(0.9953493326\)
\(L(2)\) 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 \( 1 + (3.11 + 6.26i)T \)
good3 \( 1 + (-1.11 - 1.11i)T + 9iT^{2} \)
5 \( 1 + (5.47 - 5.47i)T - 25iT^{2} \)
11 \( 1 + (2.57 + 2.57i)T + 121iT^{2} \)
13 \( 1 + (-11.1 - 11.1i)T + 169iT^{2} \)
17 \( 1 + 24.6iT - 289T^{2} \)
19 \( 1 + (10.9 + 10.9i)T + 361iT^{2} \)
23 \( 1 - 10.3iT - 529T^{2} \)
29 \( 1 + (-24.5 + 24.5i)T - 841iT^{2} \)
31 \( 1 + 14.5iT - 961T^{2} \)
37 \( 1 + (2.55 + 2.55i)T + 1.36e3iT^{2} \)
41 \( 1 - 48.3T + 1.68e3T^{2} \)
43 \( 1 + (46.0 + 46.0i)T + 1.84e3iT^{2} \)
47 \( 1 + 19.3iT - 2.20e3T^{2} \)
53 \( 1 + (-8.08 - 8.08i)T + 2.80e3iT^{2} \)
59 \( 1 + (61.0 - 61.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (75.6 + 75.6i)T + 3.72e3iT^{2} \)
67 \( 1 + (-81.1 + 81.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 9.46iT - 5.04e3T^{2} \)
73 \( 1 + 36.7T + 5.32e3T^{2} \)
79 \( 1 + 55.0T + 6.24e3T^{2} \)
83 \( 1 + (-42.2 - 42.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 8.11T + 7.92e3T^{2} \)
97 \( 1 + 141. iT - 9.40e3T^{2} \)
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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

−10.81750772473518178743963212308, −9.815604997977110972534982707968, −8.970075112590164197633411287885, −7.83262331506915123562140490306, −6.97003753474745386258639837874, −6.35354394050254177361087374342, −4.43602059811627739643352797983, −3.68626498419876827232708345399, −2.86159787439657065953497243587, −0.41714031230306430197552865835, 1.45264324363797220266584945166, 3.00944897226377011876117242397, 4.23456221241975681144418966156, 5.30689872086658816120296711795, 6.36463289260439234107270175512, 7.82926165171677489795664081395, 8.348834806310678028395323906112, 8.820862770648473419812454280038, 10.26514967758725191716523227409, 11.08698031595786450764434181490

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