Properties

Label 2-448-8.5-c1-0-8
Degree $2$
Conductor $448$
Sign $0.707 + 0.707i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·3-s − 4i·5-s − 7-s − 9-s − 2i·11-s − 4i·13-s + 8·15-s + 2·17-s − 6i·19-s − 2i·21-s − 11·25-s + 4i·27-s + 8i·29-s + 8·31-s + 4·33-s + ⋯
L(s)  = 1  + 1.15i·3-s − 1.78i·5-s − 0.377·7-s − 0.333·9-s − 0.603i·11-s − 1.10i·13-s + 2.06·15-s + 0.485·17-s − 1.37i·19-s − 0.436i·21-s − 2.20·25-s + 0.769i·27-s + 1.48i·29-s + 1.43·31-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18160 - 0.489436i\)
\(L(\frac12)\) \(\approx\) \(1.18160 - 0.489436i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 + 4iT - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.79551282548855389627948135389, −10.00078034210602862699336737032, −9.078664817954983034676137389722, −8.697769018473295927210536655281, −7.52069612891295355221778290769, −5.84301046069638269995488748734, −5.07135260317979897949845342147, −4.33115154610660680330821545485, −3.12865799059035804272392854294, −0.848289605798021387670017183136, 1.83048307998533227420738297881, 2.88507441518943687830078845100, 4.17607887309210232108380887483, 6.22330473762396035709990404524, 6.47455121033329918206541337099, 7.44222462391731258193780523312, 8.033104611333762006890392439678, 9.785440064046677561822638182324, 10.14845594714283680709060130124, 11.46956331397052785848454212106

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