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Properties

Label 2-448-448.125-c0-0-0
Degree $2$
Conductor $448$
Sign $0.290 + 0.956i$
Analytic cond. $0.223581$
Root an. cond. $0.472843$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)8-s + (−0.923 − 0.382i)9-s + (1.08 − 0.216i)11-s + (−0.707 − 0.707i)14-s − i·16-s + i·18-s + (−0.617 − 0.923i)22-s + (0.707 − 1.70i)23-s + (0.382 + 0.923i)25-s + (−0.382 + 0.923i)28-s + (−1.63 − 0.324i)29-s + (−0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)8-s + (−0.923 − 0.382i)9-s + (1.08 − 0.216i)11-s + (−0.707 − 0.707i)14-s − i·16-s + i·18-s + (−0.617 − 0.923i)22-s + (0.707 − 1.70i)23-s + (0.382 + 0.923i)25-s + (−0.382 + 0.923i)28-s + (−1.63 − 0.324i)29-s + (−0.923 + 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(0.223581\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :0),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7217072909\)
\(L(\frac12)\) \(\approx\) \(0.7217072909\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
good3 \( 1 + (0.923 + 0.382i)T^{2} \)
5 \( 1 + (-0.382 - 0.923i)T^{2} \)
11 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (-0.382 + 0.923i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.382 - 0.923i)T^{2} \)
23 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2} \)
59 \( 1 + (-0.382 - 0.923i)T^{2} \)
61 \( 1 + (0.923 + 0.382i)T^{2} \)
67 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
71 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
83 \( 1 + (0.382 - 0.923i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.30535608220783449927175487973, −10.42393094501860562727785627738, −9.227819817014893372036621675043, −8.695010867257894089787807378573, −7.77083985009566780699685828694, −6.56365523694485657660693774383, −5.12654783881691722145337481504, −4.07931343259714838553717245036, −2.94726948009999986478979072023, −1.37792746542501400546364602219, 1.80633320806324638069443438443, 3.85016667805516508285764558040, 5.17860359059096523011820989178, 5.75561863199073584781311589497, 7.04119703829106614924514546309, 7.80566665899997133383930424655, 8.892752301194280483722105221444, 9.199355743121946164231162082666, 10.62154583506694796293791672668, 11.33776025087097722817306897302

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