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Properties

Label 1-349-349.143-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.426 + 0.904i$
Analytic cond. $1.62074$
Root an. cond. $1.62074$
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  + (0.530 − 0.847i)2-s + (0.403 − 0.915i)3-s + (−0.436 − 0.899i)4-s + (−0.983 + 0.179i)5-s + (−0.561 − 0.827i)6-s + (0.197 + 0.980i)7-s + (−0.994 − 0.108i)8-s + (−0.674 − 0.738i)9-s + (−0.370 + 0.928i)10-s + (−0.725 − 0.687i)11-s + (−0.999 + 0.0361i)12-s + (−0.773 + 0.633i)13-s + (0.935 + 0.353i)14-s + (−0.232 + 0.972i)15-s + (−0.619 + 0.785i)16-s + (−0.994 + 0.108i)17-s + ⋯
L(s)  = 1  + (0.530 − 0.847i)2-s + (0.403 − 0.915i)3-s + (−0.436 − 0.899i)4-s + (−0.983 + 0.179i)5-s + (−0.561 − 0.827i)6-s + (0.197 + 0.980i)7-s + (−0.994 − 0.108i)8-s + (−0.674 − 0.738i)9-s + (−0.370 + 0.928i)10-s + (−0.725 − 0.687i)11-s + (−0.999 + 0.0361i)12-s + (−0.773 + 0.633i)13-s + (0.935 + 0.353i)14-s + (−0.232 + 0.972i)15-s + (−0.619 + 0.785i)16-s + (−0.994 + 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2644185318 - 0.4168008046i\)
\(L(\frac12)\) \(\approx\) \(-0.2644185318 - 0.4168008046i\)
\(L(1)\) \(\approx\) \(0.5837847502 - 0.6436099639i\)
\(L(1)\) \(\approx\) \(0.5837847502 - 0.6436099639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.530 - 0.847i)T \)
3 \( 1 + (0.403 - 0.915i)T \)
5 \( 1 + (-0.983 + 0.179i)T \)
7 \( 1 + (0.197 + 0.980i)T \)
11 \( 1 + (-0.725 - 0.687i)T \)
13 \( 1 + (-0.773 + 0.633i)T \)
17 \( 1 + (-0.994 + 0.108i)T \)
19 \( 1 + (-0.0180 - 0.999i)T \)
23 \( 1 + (-0.773 + 0.633i)T \)
29 \( 1 + (-0.999 - 0.0361i)T \)
31 \( 1 + (0.796 - 0.605i)T \)
37 \( 1 + (-0.161 - 0.986i)T \)
41 \( 1 + (-0.994 - 0.108i)T \)
43 \( 1 + (0.958 - 0.284i)T \)
47 \( 1 + (0.907 - 0.419i)T \)
53 \( 1 + (0.0541 + 0.998i)T \)
59 \( 1 + (0.403 - 0.915i)T \)
61 \( 1 + (0.468 - 0.883i)T \)
67 \( 1 + (-0.856 - 0.515i)T \)
71 \( 1 + (0.403 + 0.915i)T \)
73 \( 1 + (0.750 - 0.661i)T \)
79 \( 1 + (0.0541 - 0.998i)T \)
83 \( 1 + (0.336 - 0.941i)T \)
89 \( 1 + (-0.968 - 0.250i)T \)
97 \( 1 + (-0.983 - 0.179i)T \)
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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

−25.51773917759957715902503821909, −24.41783579675694974332428097908, −23.71207367407842038189887830342, −22.62334073420602326954505822496, −22.4088377192427517877464743021, −20.81575370053753364118722415809, −20.48708916787934947307038373872, −19.52841519361895430764819361130, −18.05975638043779080901845020177, −17.0230658040413048404928023746, −16.322780965241886567352296353, −15.451344212839873261911595333369, −14.90808881816974729266977102488, −13.97766720752926907049710235640, −12.95993005697990746868053673033, −11.95769388081860609400793678849, −10.71402990789046001048734394122, −9.81192911173971983577178157753, −8.407807721604712328495610646088, −7.853756223970927994026269466032, −6.9659306697597634395600284561, −5.3205787881858519239723064542, −4.476585706644499834277549023148, −3.8930002702403838480488197099, −2.69103212669318356497856410964, 0.22534428249764116190164401085, 2.07865291704209288746990597344, 2.72182377276168769788968797404, 3.9075906562971644881579195884, 5.16607241743853255211608931709, 6.28442376469809795584925896807, 7.472221959525251908650862308610, 8.57712455608238917939395574246, 9.31488675153527788535459977886, 10.96775435601821455919463825715, 11.636714690899000817965529204089, 12.318984613886069015536895035944, 13.23980776427329104299611411450, 14.10551149842709125061736471164, 15.13722466034452905092890718834, 15.684183395997906966746929036685, 17.50975221401525183359558492244, 18.50847980278566884589854916934, 19.03542022043297450358191785336, 19.69785708686593014935335121073, 20.55197642529350110859960294389, 21.70739778482816358763753840566, 22.33679079264407660338260462988, 23.561683013631274325484985493111, 24.10379170142084395700491721162

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