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Properties

Label 1-3381-3381.740-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.793 - 0.608i$
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  + (0.403 − 0.915i)2-s + (−0.675 − 0.737i)4-s + (−0.275 − 0.961i)5-s + (−0.947 + 0.320i)8-s + (−0.990 − 0.135i)10-s + (0.0339 − 0.999i)11-s + (−0.488 + 0.872i)13-s + (−0.0882 + 0.996i)16-s + (0.976 + 0.215i)17-s + (0.786 − 0.618i)19-s + (−0.523 + 0.852i)20-s + (−0.900 − 0.433i)22-s + (−0.848 + 0.529i)25-s + (0.601 + 0.798i)26-s + (0.301 + 0.953i)29-s + ⋯
L(s)  = 1  + (0.403 − 0.915i)2-s + (−0.675 − 0.737i)4-s + (−0.275 − 0.961i)5-s + (−0.947 + 0.320i)8-s + (−0.990 − 0.135i)10-s + (0.0339 − 0.999i)11-s + (−0.488 + 0.872i)13-s + (−0.0882 + 0.996i)16-s + (0.976 + 0.215i)17-s + (0.786 − 0.618i)19-s + (−0.523 + 0.852i)20-s + (−0.900 − 0.433i)22-s + (−0.848 + 0.529i)25-s + (0.601 + 0.798i)26-s + (0.301 + 0.953i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.793 - 0.608i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (740, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.793 - 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6093742384 - 1.796725295i\)
\(L(\frac12)\) \(\approx\) \(0.6093742384 - 1.796725295i\)
\(L(1)\) \(\approx\) \(0.8890728688 - 0.8295549670i\)
\(L(1)\) \(\approx\) \(0.8890728688 - 0.8295549670i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.403 - 0.915i)T \)
5 \( 1 + (-0.275 - 0.961i)T \)
11 \( 1 + (0.0339 - 0.999i)T \)
13 \( 1 + (-0.488 + 0.872i)T \)
17 \( 1 + (0.976 + 0.215i)T \)
19 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (0.301 + 0.953i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (0.855 - 0.517i)T \)
41 \( 1 + (0.970 + 0.242i)T \)
43 \( 1 + (0.947 + 0.320i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.634 - 0.773i)T \)
59 \( 1 + (0.990 + 0.135i)T \)
61 \( 1 + (-0.390 - 0.920i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (-0.917 + 0.396i)T \)
73 \( 1 + (-0.155 + 0.987i)T \)
79 \( 1 + (0.235 + 0.971i)T \)
83 \( 1 + (-0.979 - 0.202i)T \)
89 \( 1 + (-0.248 + 0.968i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−18.9135037839541897726246130900, −18.20953321524025591080700065110, −17.64925799822710076710114687521, −17.09731847408086062199021691900, −16.103696671618486310720575450470, −15.57182413019152678869641311729, −14.89873539513345989911404523752, −14.43712058185319139572359796591, −13.78076608291155991467563450770, −12.92827545104386606806646014761, −12.03294202326683360304986941703, −11.836858509648994838611171347704, −10.40684135221940213846286413626, −10.0127774279234771647290879828, −9.19529029330907384731644671199, −8.03969386735568450473939851624, −7.54371924573988221872601023202, −7.16884355015957286012631852065, −6.09015597127245968951575176027, −5.63996349584734578043048493692, −4.62261396706404319706379985970, −3.97088594586253597435945268427, −3.011274893038078547815363950246, −2.51899539726248383450947969470, −0.89982493335123055996809442169, 0.694518260828546762084951984799, 1.223321073844607242309351199, 2.35040341326991919278353975894, 3.16068450395271824471125045038, 4.01162235474799435487145699385, 4.61900500275560818255718281543, 5.45141193071718766952506698544, 5.966956404909535465123206031024, 7.15557073280999030009387858020, 8.11491761768887374887537992084, 8.82671950804116732955069543016, 9.43699368031058068176827413449, 10.05715195355168999377893233046, 11.16984986142962423175063283177, 11.50882155689099335580422142559, 12.33092225405166832970484748968, 12.77220733264004209552531868545, 13.70888848677368875875599606164, 14.09565586077002454032998778894, 14.908590160999502578555492584930, 15.88217895850803073659893821009, 16.438247465255953964606436308178, 17.160415474162530204781873073298, 18.00568635061682711834921462520, 18.8151023684065527119493074371

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