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Properties

Label 1-3381-3381.158-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.765 + 0.642i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.209 − 0.977i)2-s + (−0.912 + 0.409i)4-s + (−0.568 + 0.822i)5-s + (0.591 + 0.806i)8-s + (0.923 + 0.384i)10-s + (−0.634 − 0.773i)11-s + (−0.768 − 0.639i)13-s + (0.665 − 0.746i)16-s + (0.810 − 0.585i)17-s + (−0.0475 − 0.998i)19-s + (0.182 − 0.983i)20-s + (−0.623 + 0.781i)22-s + (−0.352 − 0.935i)25-s + (−0.464 + 0.885i)26-s + (−0.101 − 0.994i)29-s + ⋯
L(s)  = 1  + (−0.209 − 0.977i)2-s + (−0.912 + 0.409i)4-s + (−0.568 + 0.822i)5-s + (0.591 + 0.806i)8-s + (0.923 + 0.384i)10-s + (−0.634 − 0.773i)11-s + (−0.768 − 0.639i)13-s + (0.665 − 0.746i)16-s + (0.810 − 0.585i)17-s + (−0.0475 − 0.998i)19-s + (0.182 − 0.983i)20-s + (−0.623 + 0.781i)22-s + (−0.352 − 0.935i)25-s + (−0.464 + 0.885i)26-s + (−0.101 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.765 + 0.642i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.765 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1028002006 - 0.2823453967i\)
\(L(\frac12)\) \(\approx\) \(-0.1028002006 - 0.2823453967i\)
\(L(1)\) \(\approx\) \(0.5836558095 - 0.3045744072i\)
\(L(1)\) \(\approx\) \(0.5836558095 - 0.3045744072i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.209 - 0.977i)T \)
5 \( 1 + (-0.568 + 0.822i)T \)
11 \( 1 + (-0.634 - 0.773i)T \)
13 \( 1 + (-0.768 - 0.639i)T \)
17 \( 1 + (0.810 - 0.585i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (-0.101 - 0.994i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (0.00679 + 0.999i)T \)
41 \( 1 + (-0.996 - 0.0815i)T \)
43 \( 1 + (0.591 - 0.806i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (-0.546 + 0.837i)T \)
59 \( 1 + (0.923 + 0.384i)T \)
61 \( 1 + (-0.534 + 0.844i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (0.377 - 0.925i)T \)
73 \( 1 + (0.601 + 0.798i)T \)
79 \( 1 + (0.995 + 0.0950i)T \)
83 \( 1 + (0.557 - 0.830i)T \)
89 \( 1 + (-0.966 - 0.255i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−19.21396810914274061987571358770, −18.472134462975939711193979563187, −17.70970427793910000499674943882, −16.969771035540849151093860114752, −16.503064148716563777139494838727, −15.91938253302347367064654357194, −15.1091132378955755433427604909, −14.620114671125478731938388221787, −13.879376528014407777570485045737, −12.85760507471093853887836471782, −12.51399530304081802580292174103, −11.76918361957955381362350964484, −10.5767510153732413519567980804, −9.87234643187744076020214455422, −9.32231963621873379900832318182, −8.36118982506946145642685242100, −7.92742122122206918540000838911, −7.25851501398851981180346528786, −6.475977605428928035127169453719, −5.42692264214481712148835121932, −5.00302478711448014705012289805, −4.20558836768621839968074991698, −3.50094049343065491741173156282, −2.02978836064252825578953846031, −1.16342195351908405461569044192, 0.12148424038724120005892863162, 0.99454024529157204030487429696, 2.453012683523308190662082148, 2.84946149713536747565296165770, 3.47459728047070721013152559578, 4.507583395778142988195857401659, 5.16822235466693069688218890661, 6.135724985791686193428383955986, 7.248801046554180696140371251920, 7.83415076130224676283258599444, 8.42819225788337228133461455830, 9.425655255022508713019560831649, 10.16661259180417274099211605133, 10.64111320582544083478382459463, 11.41727586366091845859534156497, 11.93997379785054403174725512222, 12.62758322629471711018188247259, 13.6937250289399088487144977464, 13.86018550894784336957505526314, 15.0051935109794859128018577200, 15.480704399712976703925728661033, 16.43524264914114928089868253537, 17.25635570814242819263265163830, 17.88216932151485869351577853215, 18.62539529102106697013729524860

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