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Properties

Label 2-448-8.5-c1-0-10
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

\(L(1)\) \(\approx\) \(0.539694 - 1.30293i\)
\(L(\frac12)\) \(\approx\) \(0.539694 - 1.30293i\)
\(L(\frac{3}{2})\) 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

−10.84234862107421210772298089481, −9.692735355160042060976254594716, −8.769999597219954273262648883388, −7.88206734078082528902611033174, −7.41913054658382121179675677795, −5.84297627141452938369461742564, −5.19261777527011195033390395508, −3.93215571448427892424507521756, −1.91238908325028669385014975086, −0.956466789194662675944993508567, 2.45664355227012571966901921027, 3.56117440914938672124843321079, 4.45466749713552847990957100100, 5.79563452555758540288500037805, 6.81219527801251826774398447888, 7.63325150493208951855486452931, 9.040463568769135646168134741982, 9.745001123599614608333287651050, 10.69912550489415584227879299511, 11.10340427701083350311835465238

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