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Properties

Label 1-1248-1248.563-r0-0-0
Degree $1$
Conductor $1248$
Sign $0.902 - 0.430i$
Analytic cond. $5.79568$
Root an. cond. $5.79568$
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  + (0.707 − 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.965 − 0.258i)11-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.866 − 0.5i)23-s − i·25-s + (0.258 − 0.965i)29-s + 31-s + (0.965 − 0.258i)35-s + (0.965 + 0.258i)37-s + (0.866 − 0.5i)41-s + (0.258 + 0.965i)43-s + 47-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.965 − 0.258i)11-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.866 − 0.5i)23-s − i·25-s + (0.258 − 0.965i)29-s + 31-s + (0.965 − 0.258i)35-s + (0.965 + 0.258i)37-s + (0.866 − 0.5i)41-s + (0.258 + 0.965i)43-s + 47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(5.79568\)
Root analytic conductor: \(5.79568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1248,\ (0:\ ),\ 0.902 - 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.805275524 - 0.4087448275i\)
\(L(\frac12)\) \(\approx\) \(1.805275524 - 0.4087448275i\)
\(L(1)\) \(\approx\) \(1.270749835 - 0.1287065369i\)
\(L(1)\) \(\approx\) \(1.270749835 - 0.1287065369i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (0.965 - 0.258i)T \)
67 \( 1 + (-0.258 + 0.965i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−21.18963509096912231884343123092, −20.54436670607832578443981317514, −19.6354050559032583283904786611, −18.64592616301351552202975682366, −18.037096334361416837336454591263, −17.474176611031862689442781160043, −16.72259728345561216491488122170, −15.5721057670954460921365627025, −14.97461242798617501927593361270, −14.13233130733108688802224976128, −13.50832504157511308132506480348, −12.796006305238075167654178826703, −11.585413544611261185596515692872, −10.816412542745601728858324155068, −10.398824352874623445843421052074, −9.37475406944508698178857982028, −8.501477736702032395363591469914, −7.4173414175608584815159500315, −6.974445122856169432957724749181, −5.83451568225449282653003934089, −5.005743704671557137323329478999, −4.21311921303517389663431339255, −2.82252134290281589951066178537, −2.28184718442143296478443342972, −1.04977275529250071615153875227, 0.88282972976882917570300509628, 2.06698063806977777867560108028, 2.62481260180578904789975307876, 4.25005317381050624045358309530, 4.84382712691928964875736736114, 5.77635322164697090141890249746, 6.37160416679595019213622214596, 7.81150802783054938054356494109, 8.40914955471471058970148524882, 9.033582785783390791057100771765, 10.13005651915859731746466460377, 10.79143749791757939849601213868, 11.68208881441561083739937975350, 12.72522106055169764287541635237, 13.10723273436652869105386107199, 14.06572023878026323526902956008, 14.91181414074372583128179939260, 15.58890406986336854009027887725, 16.53503698065929608289486479200, 17.35026329799319956166470202364, 17.79087086510942375990177781031, 18.73484022318680656415485592781, 19.420286704274363174300074504262, 20.57469442014811185626250339298, 21.11521315751801061078914795709

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