↑

Properties

Label 2-210-35.12-c1-0-5
Degree $2$
Conductor $210$
Sign $0.0280 + 0.999i$
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 + (−0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + (−0.851 − 2.06i)5-s − i·6-s + (−1.86 − 1.87i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.35 + 1.77i)10-s + (−2.74 − 4.75i)11-s + (0.258 + 0.965i)12-s + (2.41 + 2.41i)13-s + (2.28 + 1.33i)14-s + (2.21 − 0.286i)15-s + (0.500 − 0.866i)16-s + (−2.04 − 0.548i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.380 − 0.924i)5-s − 0.408i·6-s + (−0.704 − 0.709i)7-s + (−0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.429 + 0.561i)10-s + (−0.827 − 1.43i)11-s + (0.0747 + 0.278i)12-s + (0.670 + 0.670i)13-s + (0.611 + 0.355i)14-s + (0.572 − 0.0740i)15-s + (0.125 − 0.216i)16-s + (−0.496 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0280 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0280 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0280 + 0.999i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.0280 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387170 - 0.376451i\)
\(L(\frac12)\) \(\approx\) \(0.387170 - 0.376451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.851 + 2.06i)T \)
7 \( 1 + (1.86 + 1.87i)T \)
good11 \( 1 + (2.74 + 4.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.41 - 2.41i)T + 13iT^{2} \)
17 \( 1 + (2.04 + 0.548i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.49 + 6.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.454 + 1.69i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.684iT - 29T^{2} \)
31 \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.46 - 2.53i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.50iT - 41T^{2} \)
43 \( 1 + (1.95 - 1.95i)T - 43iT^{2} \)
47 \( 1 + (0.912 + 3.40i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.08 - 2.43i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.08 - 8.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.01 - 0.585i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.57 + 9.61i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (-1.26 + 4.70i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.21 + 4.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.05 - 4.05i)T + 83iT^{2} \)
89 \( 1 + (-3.59 + 6.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.1 + 13.1i)T - 97iT^{2} \)
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

−11.79569026797892837101507322077, −11.06304951907921995480407109199, −10.12575056445723559067434551443, −9.013229757505220466861394192213, −8.462465568423078309013808342015, −7.13774196824076960462724254266, −5.92384044019628353400478906841, −4.65679206252965106411038589122, −3.25054712896079769835694908080, −0.56939563453193576153919577584, 2.18939267973319333472040331577, 3.44427963987781838015509396645, 5.58980500578293747304733917146, 6.73509695415101672933855284751, 7.56809880238361658341639080941, 8.510236314139107317278942837954, 9.950313211841584735146797041832, 10.44830358724117108353416766485, 11.73415826088629202813376464589, 12.36904090902089452766697498995

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