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Properties

Label 2-448-16.5-c1-0-6
Degree $2$
Conductor $448$
Sign $0.998 - 0.0529i$
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  + (0.599 − 0.599i)3-s + (0.974 + 0.974i)5-s − i·7-s + 2.28i·9-s + (1.72 + 1.72i)11-s + (1.90 − 1.90i)13-s + 1.16·15-s + 6.71·17-s + (−2.94 + 2.94i)19-s + (−0.599 − 0.599i)21-s − 5.29i·23-s − 3.09i·25-s + (3.16 + 3.16i)27-s + (−3.03 + 3.03i)29-s − 1.19·31-s + ⋯
L(s)  = 1  + (0.346 − 0.346i)3-s + (0.436 + 0.436i)5-s − 0.377i·7-s + 0.760i·9-s + (0.519 + 0.519i)11-s + (0.528 − 0.528i)13-s + 0.302·15-s + 1.62·17-s + (−0.676 + 0.676i)19-s + (−0.130 − 0.130i)21-s − 1.10i·23-s − 0.619i·25-s + (0.609 + 0.609i)27-s + (−0.562 + 0.562i)29-s − 0.215·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.998 - 0.0529i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.998 - 0.0529i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75965 + 0.0465772i\)
\(L(\frac12)\) \(\approx\) \(1.75965 + 0.0465772i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + (-0.599 + 0.599i)T - 3iT^{2} \)
5 \( 1 + (-0.974 - 0.974i)T + 5iT^{2} \)
11 \( 1 + (-1.72 - 1.72i)T + 11iT^{2} \)
13 \( 1 + (-1.90 + 1.90i)T - 13iT^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 + (2.94 - 2.94i)T - 19iT^{2} \)
23 \( 1 + 5.29iT - 23T^{2} \)
29 \( 1 + (3.03 - 3.03i)T - 29iT^{2} \)
31 \( 1 + 1.19T + 31T^{2} \)
37 \( 1 + (2.25 + 2.25i)T + 37iT^{2} \)
41 \( 1 + 3.94iT - 41T^{2} \)
43 \( 1 + (-7.02 - 7.02i)T + 43iT^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + (3.01 + 3.01i)T + 53iT^{2} \)
59 \( 1 + (4.96 + 4.96i)T + 59iT^{2} \)
61 \( 1 + (9.69 - 9.69i)T - 61iT^{2} \)
67 \( 1 + (3.55 - 3.55i)T - 67iT^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 4.06T + 79T^{2} \)
83 \( 1 + (9.17 - 9.17i)T - 83iT^{2} \)
89 \( 1 - 16.9iT - 89T^{2} \)
97 \( 1 + 2.51T + 97T^{2} \)
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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

−10.70904399524421913949630789800, −10.46423973586680832315912872622, −9.341278994276162145157304403076, −8.197081773591734871520718438509, −7.53892910850498553707750741894, −6.48936638855593849893022801994, −5.51477880877495013457408000505, −4.16517574372851706354660863325, −2.88237092177036104249975016927, −1.57408314265132112678488223566, 1.38829967409650814318097357959, 3.14143075440598844888006641594, 4.08815010309051077270479885124, 5.49632188230299741293788067426, 6.21588905342594179842688158757, 7.47776873301908574593088145037, 8.716245092085372408313629360624, 9.209580209634524894151243637055, 9.932053585528926485960964934437, 11.16965587178018931566485422895

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