Properties

Label 2-210-105.23-c1-0-10
Degree $2$
Conductor $210$
Sign $0.283 + 0.959i$
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  + (−0.965 + 0.258i)2-s + (0.378 − 1.69i)3-s + (0.866 − 0.499i)4-s + (2.16 − 0.564i)5-s + (0.0717 + 1.73i)6-s + (0.113 − 2.64i)7-s + (−0.707 + 0.707i)8-s + (−2.71 − 1.27i)9-s + (−1.94 + 1.10i)10-s + (−3.54 + 2.04i)11-s + (−0.517 − 1.65i)12-s + (3.69 + 3.69i)13-s + (0.574 + 2.58i)14-s + (−0.134 − 3.87i)15-s + (0.500 − 0.866i)16-s + (1.93 − 7.20i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.218 − 0.975i)3-s + (0.433 − 0.249i)4-s + (0.967 − 0.252i)5-s + (0.0292 + 0.706i)6-s + (0.0427 − 0.999i)7-s + (−0.249 + 0.249i)8-s + (−0.904 − 0.426i)9-s + (−0.614 + 0.349i)10-s + (−1.06 + 0.616i)11-s + (−0.149 − 0.477i)12-s + (1.02 + 1.02i)13-s + (0.153 + 0.690i)14-s + (−0.0346 − 0.999i)15-s + (0.125 − 0.216i)16-s + (0.468 − 1.74i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.283 + 0.959i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.283 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.836050 - 0.624866i\)
\(L(\frac12)\) \(\approx\) \(0.836050 - 0.624866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.378 + 1.69i)T \)
5 \( 1 + (-2.16 + 0.564i)T \)
7 \( 1 + (-0.113 + 2.64i)T \)
good11 \( 1 + (3.54 - 2.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.69 - 3.69i)T + 13iT^{2} \)
17 \( 1 + (-1.93 + 7.20i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.12 + 1.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.284 + 1.06i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 + (-2.62 - 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.43 - 5.34i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.74iT - 41T^{2} \)
43 \( 1 + (-1.27 - 1.27i)T + 43iT^{2} \)
47 \( 1 + (2.97 - 0.796i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.50 - 1.47i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.09 - 8.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.814 - 1.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 0.909i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 5.89iT - 71T^{2} \)
73 \( 1 + (-1.44 + 5.38i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.18 - 4.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.31 + 6.31i)T - 83iT^{2} \)
89 \( 1 + (6.71 - 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.21 - 5.21i)T - 97iT^{2} \)
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

−12.19716569297476712048458115583, −11.09434323849981265071041097858, −10.07426101363436942598298811604, −9.155057076374770248202539641375, −8.116157228481421335124152137048, −7.09290677271143624345391750774, −6.37700388006311113438461931204, −4.92515887077122556002128678461, −2.65963224558059921494122345628, −1.21028661436690589545792907858, 2.30562816908518161634607250814, 3.49506269820413887578187008010, 5.57502837043782965822582667617, 5.98414816642692519389977395683, 8.179792003880852029882474543549, 8.578608450790374153772037667052, 9.769407869244035300965891319010, 10.53303578836534276827991321757, 11.05223708568859890024351542390, 12.58138821017170248620760804901

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