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Properties

Label 1-3407-3407.1067-r0-0-0
Degree $1$
Conductor $3407$
Sign $-0.870 + 0.492i$
Analytic cond. $15.8220$
Root an. cond. $15.8220$
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  + (−0.710 − 0.703i)2-s + (−0.0193 + 0.999i)3-s + (0.0101 + 0.999i)4-s + (−0.958 + 0.285i)5-s + (0.717 − 0.696i)6-s + (−0.919 − 0.393i)7-s + (0.696 − 0.717i)8-s + (−0.999 − 0.0387i)9-s + (0.882 + 0.471i)10-s + (−0.0230 + 0.999i)11-s + (−0.999 − 0.00922i)12-s + (−0.984 − 0.177i)13-s + (0.376 + 0.926i)14-s + (−0.266 − 0.963i)15-s + (−0.999 + 0.0202i)16-s + (0.211 − 0.977i)17-s + ⋯
L(s)  = 1  + (−0.710 − 0.703i)2-s + (−0.0193 + 0.999i)3-s + (0.0101 + 0.999i)4-s + (−0.958 + 0.285i)5-s + (0.717 − 0.696i)6-s + (−0.919 − 0.393i)7-s + (0.696 − 0.717i)8-s + (−0.999 − 0.0387i)9-s + (0.882 + 0.471i)10-s + (−0.0230 + 0.999i)11-s + (−0.999 − 0.00922i)12-s + (−0.984 − 0.177i)13-s + (0.376 + 0.926i)14-s + (−0.266 − 0.963i)15-s + (−0.999 + 0.0202i)16-s + (0.211 − 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3407\)
Sign: $-0.870 + 0.492i$
Analytic conductor: \(15.8220\)
Root analytic conductor: \(15.8220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3407} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3407,\ (0:\ ),\ -0.870 + 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08363725466 + 0.3174394995i\)
\(L(\frac12)\) \(\approx\) \(0.08363725466 + 0.3174394995i\)
\(L(1)\) \(\approx\) \(0.4767558213 + 0.08975770470i\)
\(L(1)\) \(\approx\) \(0.4767558213 + 0.08975770470i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3407 \( 1 \)
good2 \( 1 + (-0.710 - 0.703i)T \)
3 \( 1 + (-0.0193 + 0.999i)T \)
5 \( 1 + (-0.958 + 0.285i)T \)
7 \( 1 + (-0.919 - 0.393i)T \)
11 \( 1 + (-0.0230 + 0.999i)T \)
13 \( 1 + (-0.984 - 0.177i)T \)
17 \( 1 + (0.211 - 0.977i)T \)
19 \( 1 + (-0.425 + 0.904i)T \)
23 \( 1 + (0.131 - 0.991i)T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (0.880 - 0.474i)T \)
37 \( 1 + (0.102 + 0.994i)T \)
41 \( 1 + (0.493 + 0.869i)T \)
43 \( 1 + (-0.938 - 0.345i)T \)
47 \( 1 + (-0.779 + 0.626i)T \)
53 \( 1 + (0.971 - 0.237i)T \)
59 \( 1 + (0.901 - 0.431i)T \)
61 \( 1 + (0.906 + 0.421i)T \)
67 \( 1 + (0.966 + 0.255i)T \)
71 \( 1 + (-0.000922 + 0.999i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.990 - 0.137i)T \)
83 \( 1 + (0.874 + 0.484i)T \)
89 \( 1 + (0.973 - 0.226i)T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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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

−18.74750859299511398461033966478, −17.70396099036845744777114575327, −17.18525492780080129821047826033, −16.50950323785906782991410668011, −15.8640805938613605586128074321, −15.190458078491853838468919914052, −14.546006515715196781862006026672, −13.58963287560760413735322754996, −13.04759007067215724701467063766, −12.17331665546900483837772868334, −11.59280695355787651058283686304, −10.84850356642303958522831345234, −9.89475046975296219975174121385, −9.01521918164583494558354032911, −8.45847632829840391147366789243, −7.92197473739767361380628421463, −7.04835205259280489539069205122, −6.627265614431948216052234795970, −5.76932934956013206087291772133, −5.14335351775079385159073617185, −3.94303887829204036267912452992, −2.936117927751371043650203098606, −2.11478662361021083461040497517, −0.90637540597566844525072846364, −0.207058138088157128985342782079, 0.8287124417895757681149320541, 2.46491306830437859380348459860, 2.92333446910490766661951791325, 3.7249261322477701576280471152, 4.424085611516509010210999978518, 4.97968507279516526453443088152, 6.54432683654275309543621903323, 7.06419338220453006196928713303, 7.96662865126293431406214968559, 8.541913912920738350306481514987, 9.611049481373724714647790663356, 9.97641730567604817039058576743, 10.424770973159697434558096744694, 11.32244106011244255995703201989, 12.05416353056221929147123334296, 12.436812902918244976753774585306, 13.37767942525721939148247113571, 14.54187009736722658425251122181, 14.95912329910254471913057774742, 15.99640743234331161951875755756, 16.27981715538488192086839928022, 16.95994246093852136011426830587, 17.67552170682528131028848308597, 18.59713898268113525349160310767, 19.19992367150944577567848202212

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