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Properties

Label 1-448-448.181-r1-0-0
Degree $1$
Conductor $448$
Sign $0.634 + 0.773i$
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  + (−0.923 + 0.382i)3-s + (0.382 − 0.923i)5-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.382 + 0.923i)13-s + i·15-s − i·17-s + (−0.382 − 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s + 31-s + 33-s + (0.382 − 0.923i)37-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (0.382 − 0.923i)5-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.382 + 0.923i)13-s + i·15-s − i·17-s + (−0.382 − 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s + 31-s + 33-s + (0.382 − 0.923i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.634 + 0.773i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ 0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9762787205 + 0.4617454462i\)
\(L(\frac12)\) \(\approx\) \(0.9762787205 + 0.4617454462i\)
\(L(1)\) \(\approx\) \(0.7964576232 + 0.03116165246i\)
\(L(1)\) \(\approx\) \(0.7964576232 + 0.03116165246i\)

Euler product

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

−23.38978905481081796824202346055, −22.80884240193874998690231915816, −22.289625991908932394614335453702, −21.18475767896849730954765524404, −20.40311066788009270036901465832, −18.98481744863340094677600515204, −18.34246475152836793730782100709, −17.87630285629517001453171574032, −16.88495401902137298934881062462, −15.88144790808872320056277206030, −15.083220180365705844893336669389, −13.90125140208977065999489552410, −13.113705262490050368319522302795, −12.20711998161651051903495830839, −11.1985418584272539661716831801, −10.405797070362950640135838130933, −9.83281887740755894159720905758, −8.05602641064244369126261965863, −7.36545226186926013263560429349, −6.25563933261689527105381531154, −5.64163236755641901461568041380, −4.4879859918165468763451976692, −2.97369049795812060630169901083, −1.959854051526102139395979962537, −0.42749523033158600934142403832, 0.83472156403740780534176003339, 2.06956203302998503883018936634, 3.85014612999136889075260823135, 4.7004813086132855165596880746, 5.66323047940893230110367419947, 6.34047956726102128450269266458, 7.72983538007656943441628052690, 8.88816402246085989858361046486, 9.65571663309571296434932486301, 10.73667522333872797942700343430, 11.43446817554881383153450472688, 12.58619277130932223604250840036, 13.12922280469845549569571496736, 14.25918280527982931155434213722, 15.716229210110602284047975067194, 16.01221761596144768270756190418, 17.10755107847391331940714710513, 17.58141978214730511253073270080, 18.64108634251317308218177212350, 19.63865673105599861270177720890, 20.87800932119156751704390949800, 21.33538520815363141577169002459, 22.01073573649003996580405874500, 23.236288210468907547190925710337, 23.97260415850912975361165467892

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