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Properties

Label 1-385-385.179-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.121 + 0.992i$
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  + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)24-s + (0.913 − 0.406i)26-s + (−0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)24-s + (0.913 − 0.406i)26-s + (−0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.121 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1099851039 + 0.1243302432i\)
\(L(\frac12)\) \(\approx\) \(0.1099851039 + 0.1243302432i\)
\(L(1)\) \(\approx\) \(0.4352226877 - 0.1185526058i\)
\(L(1)\) \(\approx\) \(0.4352226877 - 0.1185526058i\)

Euler product

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

−24.35808373329899632453984828188, −23.38116611053395153764791010875, −22.74772544934435347942291433867, −21.94138390459960630811390727984, −20.67014511723443353789902195472, −19.85792250318091472503388760147, −18.57317230278105881625259112063, −18.037694566692326668192007269331, −17.093087111070661302219107828550, −16.51504782223951369957118149523, −15.49172427199306874489487740227, −14.98403765641281089030124622945, −13.6959538575922642556350962769, −12.53453509304494016792850790094, −11.36272563784135584344230180310, −10.55683052829097166608401673378, −9.7459314095197387145795784034, −8.82711145616385787361572682504, −7.541440093196381392584340388453, −6.77138069482638052727506594744, −5.56329735316914091919686903970, −5.09960881799673117735294774263, −3.643987151265806864400872694039, −1.69000732374758185803333881636, −0.14304552894842790683222417823, 1.45166606111283718682843105078, 2.43327932387976125864030127311, 4.0325078265933283488829676628, 4.97742655995570287946676609752, 6.529351617601418224155241553710, 7.183199389998230517480664579341, 8.42183509848019104082798064424, 9.380727373186726532443037145063, 10.50665790548794669029193324310, 11.19811028877835567009981458, 11.97338354386935940948551813779, 12.88454266356487532049485510387, 13.59881297999337271887780324077, 15.16199351300136693348720646610, 16.39972214663799516928782714792, 16.9903360883355962302445320777, 17.765716528740113012214947187068, 18.65381613761036501948832597094, 19.29412075690550122620339951881, 20.228747309164318697550919462754, 21.42765079132501311772691888535, 21.960958613105178298777295365395, 22.811177266639053761705562907024, 23.92840336380019077402294303274, 24.611721330453389445222045870837

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