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Properties

Label 1-385-385.318-r1-0-0
Degree $1$
Conductor $385$
Sign $-0.581 + 0.813i$
Analytic cond. $41.3739$
Root an. cond. $41.3739$
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  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s − i·8-s + (0.5 − 0.866i)9-s + (−0.866 − 0.5i)12-s + i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s + (0.5 − 0.866i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s − i·8-s + (0.5 − 0.866i)9-s + (−0.866 − 0.5i)12-s + i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s + (0.5 − 0.866i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(41.3739\)
Root analytic conductor: \(41.3739\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (1:\ ),\ -0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1906412739 + 0.3705085954i\)
\(L(\frac12)\) \(\approx\) \(0.1906412739 + 0.3705085954i\)
\(L(1)\) \(\approx\) \(0.5104534991 + 0.03800438941i\)
\(L(1)\) \(\approx\) \(0.5104534991 + 0.03800438941i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + iT \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + iT \)
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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

−24.325663055896023964289043067624, −23.08672340076598842085548514943, −22.69221630497707097661354377146, −21.3917193954566417948585880250, −20.13152618764985291078896809598, −19.48631411863557914541519326806, −18.25416002124360396832343922925, −17.98568751292619143573200456348, −17.07076180177784354711406133788, −16.13980466136988042912661792791, −15.528909917142105865848895227292, −14.26400535808695874672978446905, −13.23812758872363510229818946187, −12.07509812065130923254727045272, −11.23012651065347463991546826032, −10.34755047217737165157788220927, −9.49319022340414604431064902834, −8.06980993934871468230740118711, −7.54105015070781399379661041143, −6.32623610769355524860912426904, −5.72678147711747180568831634836, −4.57721381404678693288823994438, −2.57998677609989021592407677997, −1.28499965104393338193403816140, −0.21659507939507754990719682439, 1.056504960250936065799210607495, 2.42755853025251905477772954570, 3.863519561661294029060603825713, 4.7479687055207821018217335903, 6.32536917140309313051555311466, 6.961787156284900550439223781420, 8.39649118227320765615120619309, 9.29581666985041969814192055947, 10.15403915328520071086432223279, 11.01868314719199141129467861106, 11.747474626555442363656399964359, 12.53017623940528150305730300605, 13.750069167373490303874770638017, 15.24297793580041795578582296355, 16.062987301195417525746529758237, 16.73206831253788370589718982673, 17.71445282292329338376401068433, 18.18780906160940477440550976206, 19.37476277734295908547043809915, 20.14037314798810197174231668679, 21.26631151735801934405864780983, 21.750224774026254053433370180405, 22.58714721010243117860133114245, 23.80457185158364444884620665651, 24.52116963204857579196262779981

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