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Properties

Label 1-385-385.303-r0-0-0
Degree $1$
Conductor $385$
Sign $0.934 - 0.355i$
Analytic cond. $1.78793$
Root an. cond. $1.78793$
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  + (−0.406 + 0.913i)2-s + (0.207 − 0.978i)3-s + (−0.669 − 0.743i)4-s + (0.809 + 0.587i)6-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.866 + 0.5i)12-s + (0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (0.406 + 0.913i)17-s + (0.743 − 0.669i)18-s + (0.669 − 0.743i)19-s + (0.866 − 0.5i)23-s + (−0.104 − 0.994i)24-s + (−0.978 + 0.207i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (0.207 − 0.978i)3-s + (−0.669 − 0.743i)4-s + (0.809 + 0.587i)6-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.866 + 0.5i)12-s + (0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (0.406 + 0.913i)17-s + (0.743 − 0.669i)18-s + (0.669 − 0.743i)19-s + (0.866 − 0.5i)23-s + (−0.104 − 0.994i)24-s + (−0.978 + 0.207i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.934 - 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.032960835 - 0.1898321703i\)
\(L(\frac12)\) \(\approx\) \(1.032960835 - 0.1898321703i\)
\(L(1)\) \(\approx\) \(0.9107928460 + 0.005450793329i\)
\(L(1)\) \(\approx\) \(0.9107928460 + 0.005450793329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.207 + 0.978i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (0.994 - 0.104i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−24.972152314650934713652082489040, −23.03878822906700273260896008708, −22.82321085974703264318939045654, −21.60476603298266707910537692485, −21.06829518162548574272895330007, −20.21893883034525841085041753598, −19.630874923913104997848855393975, −18.40791011422129188630934588758, −17.72014152669915853700263734137, −16.532439823641384350979785354432, −15.98335979586244674448575235776, −14.68384198053054828714877809646, −13.83506711571888152255006810198, −12.80814755557759838372607768093, −11.69406555197784824318625447364, −10.89086969623664953935941266183, −10.08493204956298368569071832025, −9.281961373075024680086074970873, −8.42113499707904835025162376016, −7.43162678662095501514999954332, −5.573629281557405439812272420123, −4.69901836612417238752438830843, −3.42689385292641371516440513070, −2.91359649623246622977031671912, −1.21831373272111885548663923887, 0.86505501697994332052928122484, 2.08633496049668672949554265290, 3.73096280665698585628706304059, 5.14410336364837131989052988754, 6.25100225945387925659386965797, 6.887227388404555594354948487331, 7.91549850694152419467894043488, 8.6709666352760912360533977895, 9.567995173092198629896365047333, 10.87984296340827472665911548312, 11.94968898357549719816185081333, 13.207617588622536593167376425431, 13.74024176504416894412341015056, 14.7295468467316824150979817638, 15.5407298029526691227741450711, 16.79660850794113791459747956825, 17.30350964733737226221773964805, 18.43063022981135057073244836639, 18.89353182922214307840998453153, 19.73293740124028093716186092085, 20.815269681226322949243040631084, 22.153410708874194401898876862804, 23.117812923217231552628547638001, 23.8245522095734212463063082618, 24.428045367685063186921129925831

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