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Properties

Label 2-210-7.3-c2-0-2
Degree $2$
Conductor $210$
Sign $-0.641 - 0.766i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s − 2.44i·6-s + (6.99 − 0.325i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (−2.73 − 1.58i)10-s + (−6.09 + 10.5i)11-s + (2.99 − 1.73i)12-s + 25.3i·13-s + (5.34 + 8.33i)14-s + 3.87·15-s + (−2.00 − 3.46i)16-s + (−24.9 − 14.3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s − 0.408i·6-s + (0.998 − 0.0465i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.273 − 0.158i)10-s + (−0.554 + 0.959i)11-s + (0.249 − 0.144i)12-s + 1.95i·13-s + (0.381 + 0.595i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−1.46 − 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.641 - 0.766i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.641 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.501285 + 1.07325i\)
\(L(\frac12)\) \(\approx\) \(0.501285 + 1.07325i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (-6.99 + 0.325i)T \)
good11 \( 1 + (6.09 - 10.5i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 25.3iT - 169T^{2} \)
17 \( 1 + (24.9 + 14.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13.9 - 8.03i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-11.8 - 20.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 27.9T + 841T^{2} \)
31 \( 1 + (-20.0 - 11.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.5 - 25.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 56.9iT - 1.68e3T^{2} \)
43 \( 1 - 7.83T + 1.84e3T^{2} \)
47 \( 1 + (19.7 - 11.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (24.2 - 42.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (62.3 + 36.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-99.2 + 57.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 + 61.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 6.41T + 5.04e3T^{2} \)
73 \( 1 + (34.2 + 19.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-27.3 - 47.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 135. iT - 6.88e3T^{2} \)
89 \( 1 + (-124. + 72.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 78.9iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.39428697860764764647896458182, −11.61869328563836834977618798565, −10.88307745729664457465107549250, −9.384777887579909696233446340640, −8.262232679668601879245646892311, −7.16417052608293016487699763929, −6.57360898420384065273245506829, −4.90821505543221057463820636901, −4.35914496735298562416437316675, −2.08495543081594910551898217251, 0.63310162333397786645667231835, 2.73475786541821102652990224151, 4.29952963184732019128916221939, 5.17955675272123182621723368833, 6.28005925240611030521269822620, 8.081022224542557614984889853938, 8.670958991607291102391784231925, 10.41002538342306556985558978455, 10.83539515463937403650477710305, 11.60476379485093850196795032161

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