↑

Properties

Label 2-210-15.14-c2-0-22
Degree $2$
Conductor $210$
Sign $-0.161 + 0.986i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (−1.07 − 2.79i)3-s + 2.00·4-s + (2.52 − 4.31i)5-s + (−1.52 − 3.95i)6-s − 2.64i·7-s + 2.82·8-s + (−6.67 + 6.03i)9-s + (3.57 − 6.10i)10-s + 5.98i·11-s + (−2.15 − 5.59i)12-s − 19.0i·13-s − 3.74i·14-s + (−14.8 − 2.42i)15-s + 4.00·16-s − 2.02·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.359 − 0.933i)3-s + 0.500·4-s + (0.505 − 0.863i)5-s + (−0.253 − 0.659i)6-s − 0.377i·7-s + 0.353·8-s + (−0.742 + 0.670i)9-s + (0.357 − 0.610i)10-s + 0.544i·11-s + (−0.179 − 0.466i)12-s − 1.46i·13-s − 0.267i·14-s + (−0.986 − 0.161i)15-s + 0.250·16-s − 0.119·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.161 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36463 - 1.60615i\)
\(L(\frac12)\) \(\approx\) \(1.36463 - 1.60615i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (1.07 + 2.79i)T \)
5 \( 1 + (-2.52 + 4.31i)T \)
7 \( 1 + 2.64iT \)
good11 \( 1 - 5.98iT - 121T^{2} \)
13 \( 1 + 19.0iT - 169T^{2} \)
17 \( 1 + 2.02T + 289T^{2} \)
19 \( 1 + 21.1T + 361T^{2} \)
23 \( 1 - 32.2T + 529T^{2} \)
29 \( 1 - 12.2iT - 841T^{2} \)
31 \( 1 - 39.3T + 961T^{2} \)
37 \( 1 + 20.3iT - 1.36e3T^{2} \)
41 \( 1 - 34.7iT - 1.68e3T^{2} \)
43 \( 1 - 40.3iT - 1.84e3T^{2} \)
47 \( 1 - 72.4T + 2.20e3T^{2} \)
53 \( 1 + 57.5T + 2.80e3T^{2} \)
59 \( 1 - 71.7iT - 3.48e3T^{2} \)
61 \( 1 - 81.7T + 3.72e3T^{2} \)
67 \( 1 - 81.7iT - 4.48e3T^{2} \)
71 \( 1 + 7.70iT - 5.04e3T^{2} \)
73 \( 1 + 12.0iT - 5.32e3T^{2} \)
79 \( 1 - 131.T + 6.24e3T^{2} \)
83 \( 1 + 115.T + 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 - 8.19iT - 9.40e3T^{2} \)
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

−12.30387423131418409550563786518, −11.08436500281358785751601905965, −10.17433194061905114283681454915, −8.661591088459332340166495976647, −7.66788147554426546268757167441, −6.54537621631395943501437928328, −5.54197602990407728891671691343, −4.59394176610587621352924198848, −2.64144147783494925432335122981, −1.05511584866339343269729563167, 2.46388764134611751521001552844, 3.77304993228262359500618924867, 4.95283891634250465676566074745, 6.14124212712715677621203440320, 6.80277980182915633035624635442, 8.664773308839641864334296451917, 9.632084451159593032240164291780, 10.74805303123273020791634913699, 11.28872483385332074045678697243, 12.24883747507897307376846713700

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