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Properties

Label 1-448-448.69-r1-0-0
Degree $1$
Conductor $448$
Sign $0.0980 - 0.995i$
Analytic cond. $48.1442$
Root an. cond. $48.1442$
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)3-s + (−0.923 − 0.382i)5-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s + i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + 33-s + (−0.923 − 0.382i)37-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (−0.923 − 0.382i)5-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s + i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + 33-s + (−0.923 − 0.382i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.0980 - 0.995i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ 0.0980 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6579554681 - 0.5963360759i\)
\(L(\frac12)\) \(\approx\) \(0.6579554681 - 0.5963360759i\)
\(L(1)\) \(\approx\) \(0.6894468168 - 0.2036009330i\)
\(L(1)\) \(\approx\) \(0.6894468168 - 0.2036009330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.382 - 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.923 - 0.382i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.923 - 0.382i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.88488028971982126739306066325, −22.87497462857985096075954570667, −22.44261600947832434849811673747, −21.51960110343911836290882194105, −20.607952706525322817095769761130, −19.80571795496453009894195178965, −18.871441493227108557390714131889, −17.952145024823179524603717626923, −16.87455034127931084676965517540, −16.07399984711942816573133625577, −15.47306246284750019326506350698, −14.610336241691361899102465019095, −13.67146648189670671677108038718, −12.19593792310600679900961493468, −11.573498320989870446868420616412, −10.75504938659109087952491695224, −9.89134928445793255232321611156, −8.86067835190465376470210584668, −7.793490085694039925332509155852, −6.84443073660357724035008662638, −5.455130549701614367822659815871, −4.81971009277759581188890068823, −3.460132346260511779470415567182, −2.95334776689170616127045111540, −0.67617119052535697694743009850, 0.40693665046379546448586549811, 1.70356187973672352224155663321, 2.8915866152527300441803892790, 4.41926884318600746447235557550, 5.17568091930447622021329077326, 6.52371959782060646772305209656, 7.41466969440642642487566629044, 8.00752142793759513825536153878, 9.15523239679236593680173446434, 10.42611968181889204631258109342, 11.46653720913682794138409468592, 12.272225024849724433912730767450, 12.74125825235973005395674367908, 13.85454450295594086682517377457, 14.96199085237356946523555229582, 15.749223510002686817470031186279, 16.96305060389978043608902892069, 17.389940839650457829416727589275, 18.59149267217107697973735350854, 19.2558078380340988391149303442, 19.981354619966295144349481544312, 20.84629044931100235853973225907, 22.200468724457708854644850055575, 22.89183485222600316688994432491, 23.65491271473008615418078657505

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