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Properties

Label 1-3895-3895.69-r0-0-0
Degree $1$
Conductor $3895$
Sign $0.231 + 0.972i$
Analytic cond. $18.0883$
Root an. cond. $18.0883$
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 + (−0.965 + 0.258i)3-s + (0.978 − 0.207i)4-s + (0.933 − 0.358i)6-s + (0.156 + 0.987i)7-s + (−0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.453 − 0.891i)11-s + (−0.891 + 0.453i)12-s + (0.358 + 0.933i)13-s + (−0.258 − 0.965i)14-s + (0.913 − 0.406i)16-s + (0.838 − 0.544i)17-s + (−0.809 + 0.587i)18-s + (−0.406 − 0.913i)21-s + (−0.358 + 0.933i)22-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (−0.965 + 0.258i)3-s + (0.978 − 0.207i)4-s + (0.933 − 0.358i)6-s + (0.156 + 0.987i)7-s + (−0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.453 − 0.891i)11-s + (−0.891 + 0.453i)12-s + (0.358 + 0.933i)13-s + (−0.258 − 0.965i)14-s + (0.913 − 0.406i)16-s + (0.838 − 0.544i)17-s + (−0.809 + 0.587i)18-s + (−0.406 − 0.913i)21-s + (−0.358 + 0.933i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ 0.231 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7376302701 + 0.5825382466i\)
\(L(\frac12)\) \(\approx\) \(0.7376302701 + 0.5825382466i\)
\(L(1)\) \(\approx\) \(0.6154276176 + 0.1766258810i\)
\(L(1)\) \(\approx\) \(0.6154276176 + 0.1766258810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.156 + 0.987i)T \)
11 \( 1 + (0.453 - 0.891i)T \)
13 \( 1 + (0.358 + 0.933i)T \)
17 \( 1 + (0.838 - 0.544i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (0.838 + 0.544i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.994 - 0.104i)T \)
47 \( 1 + (-0.933 + 0.358i)T \)
53 \( 1 + (0.838 + 0.544i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.994 + 0.104i)T \)
67 \( 1 + (-0.544 + 0.838i)T \)
71 \( 1 + (0.544 + 0.838i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.777 - 0.629i)T \)
97 \( 1 + (-0.998 - 0.0523i)T \)
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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

−18.15923436784966673964704582614, −17.68024712045264944274660650580, −17.026544786073101566419990136219, −16.7996605619405645251032778627, −15.88524593631898673385758736332, −15.1929404474201990125307157921, −14.48402455051558672392552497509, −13.27955233497808032091471344978, −12.76586264892085160288558644665, −12.02197388383507400718431441551, −11.34898044149104556317564482413, −10.716380331966357530816727540940, −10.03086708268017448951282457923, −9.76199099015583469994609846129, −8.430437598314911183893740157602, −7.83891743631617124964606385424, −7.19630216304417083382519111726, −6.54909617520459017334817994557, −5.89995225097616232470074327869, −4.90267376119409977056418046952, −4.07990479734661433890333026148, −3.138263201644072887272098310089, −2.00457945040929160508300842611, −1.148132664608076206952438866846, −0.62174960491229469688316981042, 0.92120586964272113340790750725, 1.45246068904458023113658232428, 2.651544689352273280238415947324, 3.43478654352096332208968825951, 4.57868754952092562120935038036, 5.56768782063447049436689717870, 5.88702858261765264062705422308, 6.77811431346272987415263090288, 7.30629308917810733111937846682, 8.48432664275618470342666580790, 8.9604089942121608564913297498, 9.56329405461297652034284705997, 10.403506500478585873476355105281, 11.13074371456695336635435174792, 11.66132028995449572430978079546, 12.072134492984250407523336716827, 12.92722989028684690362170831974, 14.18660760539134495290807814276, 14.70505949050479226181608703480, 15.72946499617319133958335204148, 16.103672061962525252741531316681, 16.58496100558791071547046865435, 17.38843644614429448993898734359, 17.95962053063652876651475706858, 18.64606165043932508424582524705

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