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Properties

Label 2-210-21.2-c2-0-15
Degree $2$
Conductor $210$
Sign $0.564 + 0.825i$
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  + (−1.22 − 0.707i)2-s + (2.66 − 1.37i)3-s + (0.999 + 1.73i)4-s + (−1.93 − 1.11i)5-s + (−4.23 − 0.198i)6-s + (3.85 + 5.84i)7-s − 2.82i·8-s + (5.20 − 7.33i)9-s + (1.58 + 2.73i)10-s + (10.1 − 5.87i)11-s + (5.04 + 3.24i)12-s + 0.292·13-s + (−0.595 − 9.88i)14-s + (−6.70 − 0.314i)15-s + (−2.00 + 3.46i)16-s + (25.0 − 14.4i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.888 − 0.458i)3-s + (0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.706 − 0.0331i)6-s + (0.551 + 0.834i)7-s − 0.353i·8-s + (0.578 − 0.815i)9-s + (0.158 + 0.273i)10-s + (0.925 − 0.534i)11-s + (0.420 + 0.270i)12-s + 0.0224·13-s + (−0.0425 − 0.705i)14-s + (−0.446 − 0.0209i)15-s + (−0.125 + 0.216i)16-s + (1.47 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42515 - 0.751450i\)
\(L(\frac12)\) \(\approx\) \(1.42515 - 0.751450i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-2.66 + 1.37i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-3.85 - 5.84i)T \)
good11 \( 1 + (-10.1 + 5.87i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 0.292T + 169T^{2} \)
17 \( 1 + (-25.0 + 14.4i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (3.00 - 5.20i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (27.2 + 15.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 27.0iT - 841T^{2} \)
31 \( 1 + (-1.74 - 3.02i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-27.5 + 47.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 23.5iT - 1.68e3T^{2} \)
43 \( 1 - 44.4T + 1.84e3T^{2} \)
47 \( 1 + (-2.48 - 1.43i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (79.4 - 45.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (67.9 - 39.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (11.7 - 20.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-35.3 - 61.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 + (-33.8 - 58.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (65.2 - 113. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 50.5iT - 6.88e3T^{2} \)
89 \( 1 + (-15.5 - 8.99i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 32.2T + 9.40e3T^{2} \)
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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

−12.18485813069025909373094076777, −11.13282812878606413916368005799, −9.721736223599598702239007060623, −8.920563846918709912468239189378, −8.179358067660344023717473271120, −7.31933286056481322719476392374, −5.87478249083270973626754955877, −4.02350685795076541254594300425, −2.73288415330383757608148082906, −1.24213372447664903911362591440, 1.60605866885011041263444719765, 3.56019178909156969072502621797, 4.60095852700811715754887106308, 6.36325089825995236385124647940, 7.80584926596715717316164540819, 7.942444702882140915955211623371, 9.438423582342164733948658251891, 10.06758040500322169780680048794, 11.04751262975972664128417400151, 12.14704013426755297012978135529

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