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Properties

Label 1-385-385.2-r0-0-0
Degree $1$
Conductor $385$
Sign $0.848 - 0.528i$
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  + (0.994 + 0.104i)2-s + (0.743 − 0.669i)3-s + (0.978 + 0.207i)4-s + (0.809 − 0.587i)6-s + (0.951 + 0.309i)8-s + (0.104 − 0.994i)9-s + (0.866 − 0.5i)12-s + (0.587 − 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.994 + 0.104i)17-s + (0.207 − 0.978i)18-s + (−0.978 + 0.207i)19-s + (−0.866 + 0.5i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.743 − 0.669i)3-s + (0.978 + 0.207i)4-s + (0.809 − 0.587i)6-s + (0.951 + 0.309i)8-s + (0.104 − 0.994i)9-s + (0.866 − 0.5i)12-s + (0.587 − 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.994 + 0.104i)17-s + (0.207 − 0.978i)18-s + (−0.978 + 0.207i)19-s + (−0.866 + 0.5i)23-s + (0.913 − 0.406i)24-s + (0.669 − 0.743i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.848 - 0.528i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.848 - 0.528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.970062640 - 0.8491581760i\)
\(L(\frac12)\) \(\approx\) \(2.970062640 - 0.8491581760i\)
\(L(1)\) \(\approx\) \(2.284636438 - 0.3942574383i\)
\(L(1)\) \(\approx\) \(2.284636438 - 0.3942574383i\)

Euler product

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

−24.515403999002041729587798904548, −23.70815330040670159822778531314, −22.67137416040247713116842818172, −21.870299623970169323668255936402, −21.166807490308779207252316453573, −20.475298463698831627121958498253, −19.59287979119276851792265328213, −18.87014408527979076674075541236, −17.30118303863195816387756874768, −16.169464951553588368654318354692, −15.630808486394538229385145910748, −14.727687074712761889835759829199, −13.8747636976110620099569720588, −13.2900225852637448004646819090, −12.08663170021914161484233322777, −11.03568225287043383073455056025, −10.3081634807280018793480519679, −9.09806583572666388730928873688, −8.131719256991610763542820997473, −6.86711244512036971940777267331, −5.895964021403638959766132684578, −4.43576630674686042511740183138, −4.13479651628793491640543068068, −2.75389124564427136127511340742, −1.9176296928734645415990358824, 1.501159683459285118785910920322, 2.59036707090640778534544439280, 3.54817990681987863666149794949, 4.595183212213505293901961819977, 6.04473057562892683256852377485, 6.66038241064436737561313749985, 7.876483055447737836220179624270, 8.51697405936321660796627133452, 10.01354483632861356836362970480, 11.162531682007482416714031419933, 12.14386816896655461696488028341, 13.121885923963710522556825969, 13.49029354972738114716849317592, 14.64486962210448213227909911696, 15.23653623094080262262974493858, 16.177890537233854030725804911991, 17.45031124193326969720022238133, 18.322893761297448419693527092809, 19.56819296667595687157176844652, 20.06560288993220611219295808681, 20.95517254287752635999647432843, 21.803919823798632688959896657108, 22.86115260788458383110722347963, 23.64100131065849328301467081729, 24.334839240582385262746042522560

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