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Properties

Label 1-3381-3381.134-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.204 + 0.978i$
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  + (−0.262 + 0.965i)2-s + (−0.862 − 0.505i)4-s + (−0.979 + 0.202i)5-s + (0.714 − 0.699i)8-s + (0.0611 − 0.998i)10-s + (0.917 + 0.396i)11-s + (0.986 − 0.162i)13-s + (0.488 + 0.872i)16-s + (−0.862 + 0.505i)17-s + (0.142 + 0.989i)19-s + (0.947 + 0.320i)20-s + (−0.623 + 0.781i)22-s + (0.917 − 0.396i)25-s + (−0.101 + 0.994i)26-s + (0.862 − 0.505i)29-s + ⋯
L(s)  = 1  + (−0.262 + 0.965i)2-s + (−0.862 − 0.505i)4-s + (−0.979 + 0.202i)5-s + (0.714 − 0.699i)8-s + (0.0611 − 0.998i)10-s + (0.917 + 0.396i)11-s + (0.986 − 0.162i)13-s + (0.488 + 0.872i)16-s + (−0.862 + 0.505i)17-s + (0.142 + 0.989i)19-s + (0.947 + 0.320i)20-s + (−0.623 + 0.781i)22-s + (0.917 − 0.396i)25-s + (−0.101 + 0.994i)26-s + (0.862 − 0.505i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7626943945 + 0.9381851438i\)
\(L(\frac12)\) \(\approx\) \(0.7626943945 + 0.9381851438i\)
\(L(1)\) \(\approx\) \(0.7232391699 + 0.4252407861i\)
\(L(1)\) \(\approx\) \(0.7232391699 + 0.4252407861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.262 + 0.965i)T \)
5 \( 1 + (-0.979 + 0.202i)T \)
11 \( 1 + (0.917 + 0.396i)T \)
13 \( 1 + (0.986 - 0.162i)T \)
17 \( 1 + (-0.862 + 0.505i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.862 - 0.505i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.970 + 0.242i)T \)
41 \( 1 + (0.979 - 0.202i)T \)
43 \( 1 + (0.714 + 0.699i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (-0.377 + 0.925i)T \)
59 \( 1 + (0.0611 - 0.998i)T \)
61 \( 1 + (-0.101 - 0.994i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.182 - 0.983i)T \)
73 \( 1 + (-0.301 + 0.953i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (-0.768 + 0.639i)T \)
89 \( 1 + (-0.992 + 0.122i)T \)
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

−19.07679652554193171058319611109, −17.88543826603411993542290889088, −17.56028814987804268910992678327, −16.532442477547771143239200912496, −15.97544207116444083473689197120, −15.27709810422770368705703688766, −14.20134608996035502249347492225, −13.67657559149154936175946262359, −12.93789373424187434728629441830, −12.12080189068498517852091099711, −11.54717772316854061355462698726, −11.09387348791678046694218897517, −10.41203484906805082223814695940, −9.22021280967217208386297646149, −8.87711558381783955478245022130, −8.28676285392900034589448669366, −7.30616189027603945045500730471, −6.6156802419181546338244815447, −5.46329687849115959487346680675, −4.402868150878262450992556674, −4.1117429747928233722164480463, −3.200801760718303686484266414159, −2.47973058110490959824118946987, −1.244495279622272062179735267815, −0.6337617363480547613169881868, 0.8028328064474925577600139894, 1.68182123723388474359618010117, 3.14060767099568978919033531838, 4.103330794449782034216902690189, 4.32357638864193910132895953905, 5.48489053537436365261478114014, 6.43966683604758851118898747891, 6.70978883572732726610960242245, 7.73945616405593665777569686721, 8.28042411598876397214636571995, 8.86931125617926873405058339181, 9.70822631368425505923067285445, 10.59811101940474956248980097700, 11.16213382514152277953362944134, 12.21198790932182490285156062271, 12.670328530835265900338966231656, 13.88552312756871885798618913792, 14.1461143809969058879553337600, 15.135139858498507910421471333949, 15.58814997140505130558145251818, 16.08041258469965036014029256770, 16.94987048628572276281881837099, 17.543888935497264349226810720350, 18.23870670588422361527498906304, 18.99499568987966859373064579625

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