↑

Properties

Label 1-7e3-343.218-r0-0-0
Degree $1$
Conductor $343$
Sign $0.537 - 0.843i$
Analytic cond. $1.59288$
Root an. cond. $1.59288$
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  + (−0.838 + 0.545i)2-s + (0.284 − 0.958i)3-s + (0.404 − 0.914i)4-s + (0.871 + 0.490i)5-s + (0.284 + 0.958i)6-s + (0.159 + 0.987i)8-s + (−0.838 − 0.545i)9-s + (−0.997 + 0.0640i)10-s + (−0.949 − 0.315i)11-s + (−0.761 − 0.648i)12-s + (0.518 − 0.855i)13-s + (0.718 − 0.695i)15-s + (−0.672 − 0.740i)16-s + (0.801 − 0.598i)17-s + 18-s + 19-s + ⋯
L(s)  = 1  + (−0.838 + 0.545i)2-s + (0.284 − 0.958i)3-s + (0.404 − 0.914i)4-s + (0.871 + 0.490i)5-s + (0.284 + 0.958i)6-s + (0.159 + 0.987i)8-s + (−0.838 − 0.545i)9-s + (−0.997 + 0.0640i)10-s + (−0.949 − 0.315i)11-s + (−0.761 − 0.648i)12-s + (0.518 − 0.855i)13-s + (0.718 − 0.695i)15-s + (−0.672 − 0.740i)16-s + (0.801 − 0.598i)17-s + 18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(1.59288\)
Root analytic conductor: \(1.59288\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 343,\ (0:\ ),\ 0.537 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8884269022 - 0.4870401693i\)
\(L(\frac12)\) \(\approx\) \(0.8884269022 - 0.4870401693i\)
\(L(1)\) \(\approx\) \(0.8714387817 - 0.1695235401i\)
\(L(1)\) \(\approx\) \(0.8714387817 - 0.1695235401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.838 + 0.545i)T \)
3 \( 1 + (0.284 - 0.958i)T \)
5 \( 1 + (0.871 + 0.490i)T \)
11 \( 1 + (-0.949 - 0.315i)T \)
13 \( 1 + (0.518 - 0.855i)T \)
17 \( 1 + (0.801 - 0.598i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.981 - 0.191i)T \)
29 \( 1 + (-0.981 + 0.191i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (0.801 - 0.598i)T \)
41 \( 1 + (-0.572 - 0.820i)T \)
43 \( 1 + (0.159 - 0.987i)T \)
47 \( 1 + (0.718 + 0.695i)T \)
53 \( 1 + (0.967 - 0.253i)T \)
59 \( 1 + (-0.572 + 0.820i)T \)
61 \( 1 + (-0.462 + 0.886i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.981 - 0.191i)T \)
73 \( 1 + (0.718 - 0.695i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (-0.0960 - 0.995i)T \)
89 \( 1 + (0.991 + 0.127i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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

−25.48632430211362426418354886025, −24.45494868257294039071554174357, −23.12786196709874034738634328823, −21.737025712738414255358366138446, −21.474165982957134365170178573851, −20.54602039663990073767545312602, −20.0520740405288406167243179305, −18.74466981874754439470428671423, −17.9665806027772438617729423957, −16.894841332512207597693079310872, −16.33660095038220471169510240746, −15.45829245653707286234377584362, −14.08644511351164075447675615401, −13.24549445610817731496312058135, −12.08016096851664083475585269619, −11.02889192103739524389104879829, −9.982189631376902618507109683074, −9.6485054965485443841965519368, −8.57464221579941082081063382151, −7.76840571342401756824401199133, −6.143487545062318731278366546377, −4.98096352720786260678524625622, −3.778565879532795509459370385658, −2.62786369933535143212151245166, −1.51352127883065275391175356469, 0.84354460475738122719817548537, 2.14549801131841769105551300068, 3.06090466465173190051470985614, 5.641199269121819185343692972400, 5.82318870328153314967801876519, 7.25904062827413493763019347300, 7.77600033184841972383643084995, 8.87635910358286149213158670633, 9.90370030491087110236485058904, 10.74841732675352099619986911554, 11.89559161522342066421893318200, 13.300629463712128355312391363007, 13.887486994933360011110825867341, 14.81489471337982167471393747607, 15.86091357473131055825399164821, 16.93831148662358465541870685724, 17.96018965080709662368654943227, 18.335828103392204305336914543461, 18.98337389539732688196521547967, 20.304620311297938036144703283191, 20.78196036455294928460873265566, 22.39211208662268590785739247795, 23.23192978985606879316271917305, 24.21054289039793317943324883744, 24.85695337445192813968467144319

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