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Properties

Label 1-3381-3381.719-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  + (−0.994 − 0.108i)2-s + (0.976 + 0.215i)4-s + (−0.694 − 0.719i)5-s + (−0.947 − 0.320i)8-s + (0.612 + 0.790i)10-s + (0.848 − 0.529i)11-s + (−0.488 − 0.872i)13-s + (0.906 + 0.421i)16-s + (−0.675 − 0.737i)17-s + (−0.928 + 0.371i)19-s + (−0.523 − 0.852i)20-s + (−0.900 + 0.433i)22-s + (−0.0339 + 0.999i)25-s + (0.390 + 0.920i)26-s + (0.301 − 0.953i)29-s + ⋯
L(s)  = 1  + (−0.994 − 0.108i)2-s + (0.976 + 0.215i)4-s + (−0.694 − 0.719i)5-s + (−0.947 − 0.320i)8-s + (0.612 + 0.790i)10-s + (0.848 − 0.529i)11-s + (−0.488 − 0.872i)13-s + (0.906 + 0.421i)16-s + (−0.675 − 0.737i)17-s + (−0.928 + 0.371i)19-s + (−0.523 − 0.852i)20-s + (−0.900 + 0.433i)22-s + (−0.0339 + 0.999i)25-s + (0.390 + 0.920i)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: $\chi_{3381} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.793 + 0.608i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09172884354 - 0.2704602904i\)
\(L(\frac12)\) \(\approx\) \(-0.09172884354 - 0.2704602904i\)
\(L(1)\) \(\approx\) \(0.4970737326 - 0.2012554659i\)
\(L(1)\) \(\approx\) \(0.4970737326 - 0.2012554659i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.994 - 0.108i)T \)
5 \( 1 + (-0.694 - 0.719i)T \)
11 \( 1 + (0.848 - 0.529i)T \)
13 \( 1 + (-0.488 - 0.872i)T \)
17 \( 1 + (-0.675 - 0.737i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (0.301 - 0.953i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (-0.876 + 0.482i)T \)
41 \( 1 + (0.970 - 0.242i)T \)
43 \( 1 + (0.947 - 0.320i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (0.352 - 0.935i)T \)
59 \( 1 + (-0.612 - 0.790i)T \)
61 \( 1 + (-0.601 - 0.798i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.917 - 0.396i)T \)
73 \( 1 + (-0.777 + 0.628i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (-0.979 + 0.202i)T \)
89 \( 1 + (0.963 + 0.268i)T \)
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

−19.35405408704793629761700950671, −18.61355826998661495090839862213, −17.70500199780317052984298268222, −17.42035057620074479172851392257, −16.46429385581730148166383183589, −15.94828692678489769646351371068, −15.012601569901620930749889442562, −14.73901989412081007387567384888, −13.99185396585962726540027920849, −12.633569110902969306542568778906, −12.10295415892817341861184663391, −11.40142345532239864297327019267, −10.68789674148057000432968640096, −10.257172926806033432898224718099, −9.04618358375995291310539002226, −8.89161365029258751522099444831, −7.84883224876802767000052813015, −7.01486401409088424580366615315, −6.76418731904687234722201261582, −5.96473378277556565050755875164, −4.61961909512753416485194605498, −3.96399359889953720974719952569, −2.940278051400112153579580940982, −2.12868422711414472016686294843, −1.345606936893204071122977543741, 0.14684103226522186523353451396, 0.87051571069550123574065654721, 1.92625483126121785075026903050, 2.82437323390943041685624705066, 3.74593108973150995766147944578, 4.49103906967564229764396784862, 5.56617496880724344914577030226, 6.36106050537789536740244474573, 7.16016985053809863297687540422, 8.005942419253204839081880290628, 8.37789531699511292394814439482, 9.249440914006476460485003847484, 9.707957667711596392170334239720, 10.778099634435199279368945166514, 11.28997255313031254453650095270, 12.02858002308012490947681902843, 12.56376721200455229273169242673, 13.37180202761114576448553698800, 14.44878291432127379497150124504, 15.22845775145028649208696924714, 15.788510367996255343190251955369, 16.40890951503986796169039645072, 17.29432173875116666533974264732, 17.37896771524010977466414251985, 18.51771895942963176110121232785

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