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Properties

Label 1-3895-3895.1129-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: $\chi_{3895} (1129, \cdot )$
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

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

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