Properties

Label 2-448-448.237-c0-0-0
Degree $2$
Conductor $448$
Sign $0.881 + 0.471i$
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  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.382 + 0.923i)7-s + (0.382 − 0.923i)8-s + (−0.382 + 0.923i)9-s + (−1.63 − 1.08i)11-s + (0.707 + 0.707i)14-s i·16-s + i·18-s + (−1.92 − 0.382i)22-s + (−0.707 − 0.292i)23-s + (−0.923 + 0.382i)25-s + (0.923 + 0.382i)28-s + (0.324 − 0.216i)29-s + (−0.382 − 0.923i)32-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.382 + 0.923i)7-s + (0.382 − 0.923i)8-s + (−0.382 + 0.923i)9-s + (−1.63 − 1.08i)11-s + (0.707 + 0.707i)14-s i·16-s + i·18-s + (−1.92 − 0.382i)22-s + (−0.707 − 0.292i)23-s + (−0.923 + 0.382i)25-s + (0.923 + 0.382i)28-s + (0.324 − 0.216i)29-s + (−0.382 − 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(0.223581\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :0),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.348362145\)
\(L(\frac12)\) \(\approx\) \(1.348362145\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
good3 \( 1 + (0.382 - 0.923i)T^{2} \)
5 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (0.923 + 0.382i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
83 \( 1 + (-0.923 - 0.382i)T^{2} \)
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

−11.25889680765901997333165693436, −10.72074279127834943229143714834, −9.704933223970745721241592491119, −8.300174201075096143275329910796, −7.73143666031816254928227354276, −6.03750220180130946964951952416, −5.52780161358421626177057858936, −4.60794267644005067742792906239, −3.01039583882603933707881056376, −2.21115274458941097455014775859, 2.30170510873943331393055199239, 3.66916154241648517205810318620, 4.63911563441289088642065602172, 5.60546482070183577005311446829, 6.72214542204593961241395153067, 7.60475445671023742231489118347, 8.251689162258001549592734496789, 9.808936546442561759807368362338, 10.57333326349591098139342442887, 11.61228442457266119951403210222

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