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Properties

Label 2-210-105.68-c2-0-25
Degree $2$
Conductor $210$
Sign $-0.443 + 0.896i$
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  + (−1.36 + 0.366i)2-s + (1.05 − 2.80i)3-s + (1.73 − i)4-s + (−1.47 − 4.77i)5-s + (−0.414 + 4.22i)6-s + (6.99 − 0.132i)7-s + (−1.99 + 2i)8-s + (−6.77 − 5.92i)9-s + (3.76 + 5.98i)10-s + (9.30 − 5.36i)11-s + (−0.979 − 5.91i)12-s + (−3.28 + 3.28i)13-s + (−9.51 + 2.74i)14-s + (−14.9 − 0.902i)15-s + (1.99 − 3.46i)16-s + (−16.0 − 4.29i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.351 − 0.936i)3-s + (0.433 − 0.250i)4-s + (−0.295 − 0.955i)5-s + (−0.0690 + 0.703i)6-s + (0.999 − 0.0189i)7-s + (−0.249 + 0.250i)8-s + (−0.752 − 0.658i)9-s + (0.376 + 0.598i)10-s + (0.845 − 0.488i)11-s + (−0.0816 − 0.493i)12-s + (−0.252 + 0.252i)13-s + (−0.679 + 0.195i)14-s + (−0.998 − 0.0601i)15-s + (0.124 − 0.216i)16-s + (−0.942 − 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.443 + 0.896i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.443 + 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.616307 - 0.992880i\)
\(L(\frac12)\) \(\approx\) \(0.616307 - 0.992880i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 + (-1.05 + 2.80i)T \)
5 \( 1 + (1.47 + 4.77i)T \)
7 \( 1 + (-6.99 + 0.132i)T \)
good11 \( 1 + (-9.30 + 5.36i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.28 - 3.28i)T - 169iT^{2} \)
17 \( 1 + (16.0 + 4.29i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (1.04 - 1.81i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (3.40 + 12.7i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 21.8T + 841T^{2} \)
31 \( 1 + (-40.3 + 23.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (10.9 + 40.7i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 41.7T + 1.68e3T^{2} \)
43 \( 1 + (32.4 + 32.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18.0 - 67.3i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (31.2 + 8.36i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (8.21 - 4.74i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-100. - 58.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (2.25 + 0.604i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 82.6iT - 5.04e3T^{2} \)
73 \( 1 + (-89.0 - 23.8i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (27.5 + 15.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-37.0 - 37.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (-136. - 78.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (41.8 + 41.8i)T + 9.40e3iT^{2} \)
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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

−11.74009257078057489176342549496, −11.14733974760935862953454197447, −9.405790580544237244022180704390, −8.667214660198602814156645908674, −7.998923480747264656382403921801, −6.97809023567063979557606967908, −5.75685847426773989402717383571, −4.24571776742608958763227770756, −2.11278398280029997100844934712, −0.805841323219734678829534011249, 2.15572126591673746211309706560, 3.59239757145501720130468352032, 4.79217150214538844147614282455, 6.49779576404805037157206781403, 7.69268032912329409407018703309, 8.560968233926274253827976319531, 9.614102168236460565043509128543, 10.48306690325144596426383047362, 11.26631100408514250661022148083, 11.91373656414047794697167116826

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