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Properties

Label 1-6369-6369.1118-r1-0-0
Degree $1$
Conductor $6369$
Sign $-0.972 - 0.234i$
Analytic cond. $684.444$
Root an. cond. $684.444$
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  + (−0.928 + 0.370i)2-s + (0.725 − 0.688i)4-s + (−0.516 + 0.856i)5-s + (−0.358 − 0.933i)7-s + (−0.418 + 0.908i)8-s + (0.162 − 0.986i)10-s + (0.539 − 0.842i)13-s + (0.678 + 0.734i)14-s + (0.0523 − 0.998i)16-s + (0.905 − 0.424i)17-s + (−0.756 + 0.654i)19-s + (0.214 + 0.976i)20-s + (−0.831 + 0.555i)23-s + (−0.465 − 0.884i)25-s + (−0.188 + 0.982i)26-s + ⋯
L(s)  = 1  + (−0.928 + 0.370i)2-s + (0.725 − 0.688i)4-s + (−0.516 + 0.856i)5-s + (−0.358 − 0.933i)7-s + (−0.418 + 0.908i)8-s + (0.162 − 0.986i)10-s + (0.539 − 0.842i)13-s + (0.678 + 0.734i)14-s + (0.0523 − 0.998i)16-s + (0.905 − 0.424i)17-s + (−0.756 + 0.654i)19-s + (0.214 + 0.976i)20-s + (−0.831 + 0.555i)23-s + (−0.465 − 0.884i)25-s + (−0.188 + 0.982i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6369\)    =    \(3 \cdot 11 \cdot 193\)
Sign: $-0.972 - 0.234i$
Analytic conductor: \(684.444\)
Root analytic conductor: \(684.444\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6369} (1118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6369,\ (1:\ ),\ -0.972 - 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008737884968 + 0.07339853956i\)
\(L(\frac12)\) \(\approx\) \(0.008737884968 + 0.07339853956i\)
\(L(1)\) \(\approx\) \(0.5811144035 + 0.09066946713i\)
\(L(1)\) \(\approx\) \(0.5811144035 + 0.09066946713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
193 \( 1 \)
good2 \( 1 + (-0.928 + 0.370i)T \)
5 \( 1 + (-0.516 + 0.856i)T \)
7 \( 1 + (-0.358 - 0.933i)T \)
13 \( 1 + (0.539 - 0.842i)T \)
17 \( 1 + (0.905 - 0.424i)T \)
19 \( 1 + (-0.756 + 0.654i)T \)
23 \( 1 + (-0.831 + 0.555i)T \)
29 \( 1 + (0.747 - 0.664i)T \)
31 \( 1 + (0.370 + 0.928i)T \)
37 \( 1 + (-0.869 - 0.494i)T \)
41 \( 1 + (0.789 + 0.613i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.550 + 0.835i)T \)
53 \( 1 + (-0.998 - 0.0457i)T \)
59 \( 1 + (-0.284 + 0.958i)T \)
61 \( 1 + (0.996 + 0.0849i)T \)
67 \( 1 + (-0.555 - 0.831i)T \)
71 \( 1 + (0.0196 - 0.999i)T \)
73 \( 1 + (0.201 + 0.979i)T \)
79 \( 1 + (0.239 - 0.970i)T \)
83 \( 1 + (0.0130 + 0.999i)T \)
89 \( 1 + (0.634 - 0.773i)T \)
97 \( 1 + (0.969 - 0.246i)T \)
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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

−17.06306314165451282281753831825, −16.51194347136230321049281915177, −15.89493558485668534575200944833, −15.50764314748629315880394374, −14.68571929960120874612052066843, −13.68101029894169663664559949746, −12.8353823608711543737471341923, −12.34271568367423353519001013537, −11.86452456146181918791110807352, −11.2640885266868702078365455538, −10.40058990450308887611468874141, −9.72378410663942243695141118744, −9.00623618356847549384310014735, −8.51822745134316065685149030119, −8.14962993331223059675222192024, −7.148066941063522928387751345111, −6.43474680313244310447000318868, −5.76960672272733031954022250669, −4.791201234841935168234163985775, −3.94771440478549572418845814588, −3.33697935106259969256382373824, −2.31890398841791375068708159133, −1.74698877441599148025303684328, −0.806449388312080861924573989485, −0.020459003474908881035953879489, 0.79156299888335967715967227340, 1.564105937062872845183864152179, 2.70125459935230157142536041076, 3.306090626691994322743734161258, 4.03464899386654152387488608306, 5.070581953210723964453616991146, 6.17039282488344909334051416242, 6.33978199397521833570408225287, 7.31367148827702220497090132224, 7.840985312850925263927957201808, 8.18355904767200176534785829056, 9.25103977188172565212794956540, 10.09733856357011634479582133784, 10.38253631382844485086424134599, 10.926435503132141352995340401, 11.754923352116439131807373420230, 12.35809187037211908948983946584, 13.377920746640747344906481782262, 14.24250532026542208761129845685, 14.467868669263413871642432817596, 15.459163877331348717916965558008, 15.92610230652695936203862174855, 16.393348881074554116668471901459, 17.30695058972660041144032284576, 17.73691791050449266732220050789

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