Properties

Label 2-210-35.17-c1-0-5
Degree $2$
Conductor $210$
Sign $0.364 + 0.931i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 − 0.499i)4-s + (−0.126 − 2.23i)5-s i·6-s + (−2.47 − 0.942i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (2.18 + 0.455i)10-s + (1.55 − 2.69i)11-s + (0.965 + 0.258i)12-s + (−3.40 − 3.40i)13-s + (1.55 − 2.14i)14-s + (0.700 + 2.12i)15-s + (0.500 + 0.866i)16-s + (−1.37 − 5.14i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 − 0.249i)4-s + (−0.0566 − 0.998i)5-s − 0.408i·6-s + (−0.934 − 0.356i)7-s + (0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (0.692 + 0.144i)10-s + (0.468 − 0.811i)11-s + (0.278 + 0.0747i)12-s + (−0.945 − 0.945i)13-s + (0.414 − 0.572i)14-s + (0.180 + 0.548i)15-s + (0.125 + 0.216i)16-s + (−0.334 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.364 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503643 - 0.343880i\)
\(L(\frac12)\) \(\approx\) \(0.503643 - 0.343880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (0.126 + 2.23i)T \)
7 \( 1 + (2.47 + 0.942i)T \)
good11 \( 1 + (-1.55 + 2.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.40 + 3.40i)T + 13iT^{2} \)
17 \( 1 + (1.37 + 5.14i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.61 - 6.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.08 - 1.36i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.49iT - 29T^{2} \)
31 \( 1 + (7.98 + 4.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.929 - 3.47i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (3.86 - 3.86i)T - 43iT^{2} \)
47 \( 1 + (3.94 + 1.05i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.890 - 3.32i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.666 - 1.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.8 + 6.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.43 + 1.72i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (-7.95 + 2.13i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.65 - 0.957i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.97 - 8.97i)T + 83iT^{2} \)
89 \( 1 + (2.03 + 3.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.69 + 2.69i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.31735722620473101999084741896, −11.26583355352650341225883344923, −9.818923631441294393061345335659, −9.430552527016304883980717687713, −8.077444922530513671007547724257, −7.08186063875238493355963375904, −5.82076631055171216947327059339, −5.05961361567717802274034798878, −3.58629041212724973734336450085, −0.58981192959008811344652750719, 2.20397868591573535635903639929, 3.59519762217931163150889183482, 5.07196330486539237125772314292, 6.73181695911473184163817483154, 7.09942977934142549402835691025, 8.995181282170582024942965209140, 9.749488173492858774255356758023, 10.72670586078066780252928007559, 11.52679281590533596445718926255, 12.42971285679885643142828332902

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