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Properties

Label 1-3381-3381.104-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.545 + 0.838i$
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  + (0.979 + 0.202i)2-s + (0.917 + 0.396i)4-s + (0.947 + 0.320i)5-s + (0.818 + 0.574i)8-s + (0.862 + 0.505i)10-s + (−0.794 + 0.607i)11-s + (0.933 − 0.359i)13-s + (0.685 + 0.728i)16-s + (0.917 − 0.396i)17-s + (0.654 + 0.755i)19-s + (0.742 + 0.670i)20-s + (−0.900 + 0.433i)22-s + (0.794 + 0.607i)25-s + (0.986 − 0.162i)26-s + (−0.917 + 0.396i)29-s + ⋯
L(s)  = 1  + (0.979 + 0.202i)2-s + (0.917 + 0.396i)4-s + (0.947 + 0.320i)5-s + (0.818 + 0.574i)8-s + (0.862 + 0.505i)10-s + (−0.794 + 0.607i)11-s + (0.933 − 0.359i)13-s + (0.685 + 0.728i)16-s + (0.917 − 0.396i)17-s + (0.654 + 0.755i)19-s + (0.742 + 0.670i)20-s + (−0.900 + 0.433i)22-s + (0.794 + 0.607i)25-s + (0.986 − 0.162i)26-s + (−0.917 + 0.396i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.040040776 + 2.191594380i\)
\(L(\frac12)\) \(\approx\) \(4.040040776 + 2.191594380i\)
\(L(1)\) \(\approx\) \(2.345088745 + 0.6707872105i\)
\(L(1)\) \(\approx\) \(2.345088745 + 0.6707872105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.979 + 0.202i)T \)
5 \( 1 + (0.947 + 0.320i)T \)
11 \( 1 + (-0.794 + 0.607i)T \)
13 \( 1 + (0.933 - 0.359i)T \)
17 \( 1 + (0.917 - 0.396i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.917 + 0.396i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (-0.523 - 0.852i)T \)
41 \( 1 + (0.947 + 0.320i)T \)
43 \( 1 + (-0.818 + 0.574i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.301 + 0.953i)T \)
59 \( 1 + (-0.862 - 0.505i)T \)
61 \( 1 + (-0.986 - 0.162i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.0203 - 0.999i)T \)
73 \( 1 + (-0.882 + 0.470i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (-0.714 + 0.699i)T \)
89 \( 1 + (0.488 - 0.872i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−18.8041221566005556136157641586, −18.10697039937271815886751575680, −17.17592831318340563230699840624, −16.49798740563541856717994715408, −15.88364676471857161384076945290, −15.237983442411510341375613734031, −14.2630814101853873014177662940, −13.75385487031797404641944320142, −13.26791313708315497183186136418, −12.66177314165528021235893956524, −11.76568052781452171840635650844, −11.11078597333542656742134625564, −10.35729068597678800647094736886, −9.7836404104312155694590985947, −8.82582822888984447058941865993, −8.015112709817854648103725064060, −7.07630906402824933574859035202, −6.21090015062996753250396669231, −5.6869472494940387045459203076, −5.10099159595518515012669640447, −4.2325513081212839324288066628, −3.25272357599425705817206124814, −2.68734707055992228808095531112, −1.66039149437406833535494310640, −0.99248686295468415391636549058, 1.2967325467490600232709821082, 2.0252989517184766397216435795, 2.97893581904877312574871565551, 3.46575319284697737061178876188, 4.58072277370825251806705446100, 5.40062565554061546108140406728, 5.81260780191575149282150173076, 6.563430243709186107616435437241, 7.50753889028098743055476241185, 7.92824348484799665258672503373, 9.104165933164703794019827618510, 10.0138650812355656577226987987, 10.52918653817227242646386644427, 11.27647973297069256091617844256, 12.18089237631736905168833324177, 12.79867031961469046090620931101, 13.48401107758836037869987068509, 13.98286721769156013357605294361, 14.69052060836888000921352170104, 15.343000762238511170975180208242, 16.09913019524270342539326610425, 16.69596727639855567249706872839, 17.487181161054743539450054127459, 18.299156764693260866815171441597, 18.67438374531708901541925775719

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