↑

Properties

Label 1-349-349.20-r0-0-0
Degree $1$
Conductor $349$
Sign $0.170 - 0.985i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.837 − 0.546i)2-s + (−0.773 − 0.633i)3-s + (0.403 − 0.915i)4-s + (0.700 + 0.713i)5-s + (−0.994 − 0.108i)6-s + (0.530 − 0.847i)7-s + (−0.161 − 0.986i)8-s + (0.197 + 0.980i)9-s + (0.976 + 0.214i)10-s + (0.907 + 0.419i)11-s + (−0.891 + 0.452i)12-s + (0.874 + 0.484i)13-s + (−0.0180 − 0.999i)14-s + (−0.0901 − 0.995i)15-s + (−0.674 − 0.738i)16-s + (−0.161 + 0.986i)17-s + ⋯
L(s)  = 1  + (0.837 − 0.546i)2-s + (−0.773 − 0.633i)3-s + (0.403 − 0.915i)4-s + (0.700 + 0.713i)5-s + (−0.994 − 0.108i)6-s + (0.530 − 0.847i)7-s + (−0.161 − 0.986i)8-s + (0.197 + 0.980i)9-s + (0.976 + 0.214i)10-s + (0.907 + 0.419i)11-s + (−0.891 + 0.452i)12-s + (0.874 + 0.484i)13-s + (−0.0180 − 0.999i)14-s + (−0.0901 − 0.995i)15-s + (−0.674 − 0.738i)16-s + (−0.161 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.170 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.555771851 - 1.309487607i\)
\(L(\frac12)\) \(\approx\) \(1.555771851 - 1.309487607i\)
\(L(1)\) \(\approx\) \(1.424585668 - 0.7637583855i\)
\(L(1)\) \(\approx\) \(1.424585668 - 0.7637583855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.837 - 0.546i)T \)
3 \( 1 + (-0.773 - 0.633i)T \)
5 \( 1 + (0.700 + 0.713i)T \)
7 \( 1 + (0.530 - 0.847i)T \)
11 \( 1 + (0.907 + 0.419i)T \)
13 \( 1 + (0.874 + 0.484i)T \)
17 \( 1 + (-0.161 + 0.986i)T \)
19 \( 1 + (-0.232 - 0.972i)T \)
23 \( 1 + (0.874 + 0.484i)T \)
29 \( 1 + (-0.891 - 0.452i)T \)
31 \( 1 + (-0.561 - 0.827i)T \)
37 \( 1 + (-0.856 + 0.515i)T \)
41 \( 1 + (-0.161 - 0.986i)T \)
43 \( 1 + (-0.817 + 0.576i)T \)
47 \( 1 + (0.796 + 0.605i)T \)
53 \( 1 + (0.647 + 0.762i)T \)
59 \( 1 + (-0.773 - 0.633i)T \)
61 \( 1 + (0.0541 - 0.998i)T \)
67 \( 1 + (-0.725 - 0.687i)T \)
71 \( 1 + (-0.773 + 0.633i)T \)
73 \( 1 + (-0.999 - 0.0361i)T \)
79 \( 1 + (0.647 - 0.762i)T \)
83 \( 1 + (-0.968 + 0.250i)T \)
89 \( 1 + (0.989 + 0.143i)T \)
97 \( 1 + (0.700 - 0.713i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.94816083926930563051208183094, −24.20695183717165340983441933615, −23.15246372922504830387593996213, −22.38275228525517750555784824565, −21.62618832537051009077749838154, −20.94699100979570911890245257522, −20.33826894870790945786408029575, −18.3938302775281430318843504774, −17.67621807596319697290098619553, −16.65199870955328500983910605447, −16.26089954626857929943973221416, −15.14842724752420883902063205609, −14.38056805026915067350690959846, −13.27370291556209907175728878857, −12.2887186988827298538974461423, −11.63214943769116584415776738593, −10.6030886790697005571939287722, −9.0627354743386929146344013174, −8.598648561424621447778952903213, −6.88605392536270171127296658581, −5.794601975514218941709139809430, −5.40643910017623785852267079526, −4.39132861030370886920869968256, −3.233217293930182643633747006302, −1.54693809493925680754346586487, 1.338939269776698235981129816177, 2.00276726412498896900112239218, 3.63832824913486919188480170614, 4.66363317648115868164496489991, 5.86035457443484961459876112265, 6.62465485627119659569895358049, 7.33385566113330247802475560857, 9.21223456859662474564339325024, 10.5034819634545381150212797553, 11.03739868181167409175330462518, 11.74569753751232132472123422012, 13.09256928793571721821517110774, 13.542689636220727493016926878653, 14.454684717496704271922565015555, 15.380495595567827694936717310397, 16.95277796553471130503736113861, 17.42936289469677311130408010217, 18.58521248045434541406825047064, 19.278155804919863359253755905344, 20.34437994867250480468356678430, 21.36814619673162937144185304257, 22.091850498863090020375897069092, 22.858624738662807432963096902984, 23.61567477470195204272758419967, 24.28865139449629014549653665760

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