Properties

Label 1-3381-3381.107-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: $\chi_{3381} (107, \cdot )$
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

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

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