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Properties

Label 1-385-385.278-r0-0-0
Degree $1$
Conductor $385$
Sign $0.474 + 0.880i$
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  + (0.207 + 0.978i)2-s + (−0.994 + 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.309 − 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.978 − 0.207i)9-s + (0.866 − 0.5i)12-s + (−0.951 − 0.309i)13-s + (0.669 − 0.743i)16-s + (0.207 − 0.978i)17-s + (0.406 + 0.913i)18-s + (0.913 + 0.406i)19-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)24-s + (0.104 − 0.994i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.994 + 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.309 − 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.978 − 0.207i)9-s + (0.866 − 0.5i)12-s + (−0.951 − 0.309i)13-s + (0.669 − 0.743i)16-s + (0.207 − 0.978i)17-s + (0.406 + 0.913i)18-s + (0.913 + 0.406i)19-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)24-s + (0.104 − 0.994i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.474 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7513643684 + 0.4484196967i\)
\(L(\frac12)\) \(\approx\) \(0.7513643684 + 0.4484196967i\)
\(L(1)\) \(\approx\) \(0.7209512507 + 0.3613342644i\)
\(L(1)\) \(\approx\) \(0.7209512507 + 0.3613342644i\)

Euler product

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

−23.96466720445957803129333409353, −23.37798834140321808626990515873, −22.46971726574180966162479912521, −21.725556909134253447401519285834, −21.18756797216660042755939699244, −19.854248379427348767206162918697, −19.228076281546724497158398135386, −18.22876051200985963071252836250, −17.48747437089317210659429275714, −16.70317099221149867515857583622, −15.4353819924516676033010870049, −14.41762527753393701400387253757, −13.29868500569580542432717034713, −12.516207428327567174364016179963, −11.7505332713605752503789648784, −10.970519664082033231128865424, −10.058750717165180098983836860644, −9.26390721280289508700903899119, −7.81844737463191676270450107641, −6.59823262778716147796986737637, −5.416109468942031826117029445900, −4.72455077403868068590288388982, −3.54645384362771004852953290259, −2.152655681400723187337907765726, −0.93015507175821468611527342275, 0.81154980414934296595928507969, 3.00729234309225431193310939506, 4.455033000770172066782899651038, 5.149732192687483311627838950786, 6.03720065797997502159809200602, 7.10342825290946390133322696809, 7.74723249456424150085743859531, 9.26629376594542383852047897369, 9.95911891545561006362939269065, 11.25338867849637621620693863460, 12.28323652354199329355451708195, 12.957498934011788850250942205515, 14.16832361249280334458095896975, 15.00019200757514514827288786122, 16.07110730655889829450810617408, 16.5461525968126946317558112771, 17.52212591917243287954864434303, 18.15629894098321833141789576380, 19.04105688451187730383565801799, 20.509515807271449329532266116679, 21.55779162790511777264826631335, 22.3986827665826154253778050035, 22.86413996497804409800573747915, 23.79863774070959664868065159759, 24.601127001670850744025288911979

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