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Properties

Label 1-3407-3407.447-r0-0-0
Degree $1$
Conductor $3407$
Sign $-0.172 - 0.985i$
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  + (0.990 − 0.139i)2-s + (−0.681 − 0.731i)3-s + (0.960 − 0.276i)4-s + (0.999 − 0.00737i)5-s + (−0.777 − 0.629i)6-s + (0.674 − 0.738i)7-s + (0.912 − 0.408i)8-s + (−0.0709 + 0.997i)9-s + (0.989 − 0.147i)10-s + (−0.976 + 0.217i)11-s + (−0.857 − 0.514i)12-s + (0.980 − 0.194i)13-s + (0.565 − 0.825i)14-s + (−0.686 − 0.726i)15-s + (0.846 − 0.531i)16-s + (0.923 − 0.384i)17-s + ⋯
L(s)  = 1  + (0.990 − 0.139i)2-s + (−0.681 − 0.731i)3-s + (0.960 − 0.276i)4-s + (0.999 − 0.00737i)5-s + (−0.777 − 0.629i)6-s + (0.674 − 0.738i)7-s + (0.912 − 0.408i)8-s + (−0.0709 + 0.997i)9-s + (0.989 − 0.147i)10-s + (−0.976 + 0.217i)11-s + (−0.857 − 0.514i)12-s + (0.980 − 0.194i)13-s + (0.565 − 0.825i)14-s + (−0.686 − 0.726i)15-s + (0.846 − 0.531i)16-s + (0.923 − 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3407\)
Sign: $-0.172 - 0.985i$
Analytic conductor: \(15.8220\)
Root analytic conductor: \(15.8220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3407} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3407,\ (0:\ ),\ -0.172 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.465882640 - 2.934132352i\)
\(L(\frac12)\) \(\approx\) \(2.465882640 - 2.934132352i\)
\(L(1)\) \(\approx\) \(1.864426166 - 0.9739664014i\)
\(L(1)\) \(\approx\) \(1.864426166 - 0.9739664014i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3407 \( 1 \)
good2 \( 1 + (0.990 - 0.139i)T \)
3 \( 1 + (-0.681 - 0.731i)T \)
5 \( 1 + (0.999 - 0.00737i)T \)
7 \( 1 + (0.674 - 0.738i)T \)
11 \( 1 + (-0.976 + 0.217i)T \)
13 \( 1 + (0.980 - 0.194i)T \)
17 \( 1 + (0.923 - 0.384i)T \)
19 \( 1 + (0.0212 - 0.999i)T \)
23 \( 1 + (-0.875 - 0.482i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (0.373 + 0.927i)T \)
37 \( 1 + (0.826 + 0.562i)T \)
41 \( 1 + (-0.631 - 0.775i)T \)
43 \( 1 + (0.390 + 0.920i)T \)
47 \( 1 + (-0.753 + 0.657i)T \)
53 \( 1 + (0.0838 - 0.996i)T \)
59 \( 1 + (0.973 + 0.226i)T \)
61 \( 1 + (-0.969 + 0.246i)T \)
67 \( 1 + (-0.839 + 0.542i)T \)
71 \( 1 + (-0.966 + 0.257i)T \)
73 \( 1 + (0.437 + 0.899i)T \)
79 \( 1 + (0.250 - 0.968i)T \)
83 \( 1 + (-0.392 + 0.919i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (-0.191 - 0.981i)T \)
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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

−18.584379977081247913093833806284, −18.37109652372616548789478044508, −17.46358821803050967388225174163, −16.65842650946579550579630913651, −16.27103465824984381293656724187, −15.452385537764756984496524512763, −14.840641307842816923517885866698, −14.24951498644805862087521832928, −13.46549836411169205524738980569, −12.75929363632291523254099471442, −12.017994660692656042702137131179, −11.402164149321559473893827876, −10.62055305509080141994329669415, −10.17961270690002950506556114279, −9.23203348966079854191450703502, −8.28345320869031524262584594456, −7.57741557136399608799032868472, −6.25094503954526597707664807068, −5.85048489308995476103936201557, −5.48595667485210079687387240976, −4.73132476369843807951454966256, −3.80146214616629202159380443634, −3.091193559410597159725887928523, −2.042432082072636361548276952161, −1.362618787441584124720192336287, 0.861194939051877795690324479116, 1.53982323288308275668787260715, 2.34517972234969848388772202657, 3.107862840933456863140908972492, 4.37043440949146132519271466861, 5.00721453544701238181633921538, 5.6237802335975963906702801070, 6.24727598269137276174748987631, 7.01266587232550062559574080428, 7.69006151606273117852228922832, 8.41216319737425970903956969318, 9.925430147214051267936962055740, 10.37344523707045293865768568230, 11.09115263209561098807297733675, 11.60514763095013089999409531468, 12.56754145891142813344577546976, 13.177091421349730963134835206054, 13.603681116682183639192042197, 14.13981133652815837878487582249, 14.94036041547473453094991886082, 16.00642057218677194662329800465, 16.45604742975400490549045932468, 17.28549045846972920586114061264, 17.969416231399704958455198118480, 18.39306692092976487639882278358

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