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Properties

Label 2-210-21.2-c2-0-19
Degree $2$
Conductor $210$
Sign $0.691 + 0.722i$
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 + (1.81 − 2.38i)3-s + (0.999 + 1.73i)4-s + (−1.93 − 1.11i)5-s + (3.91 − 1.64i)6-s + (−2.08 − 6.68i)7-s + 2.82i·8-s + (−2.40 − 8.67i)9-s + (−1.58 − 2.73i)10-s + (14.2 − 8.25i)11-s + (5.95 + 0.757i)12-s + 13.0·13-s + (2.16 − 9.65i)14-s + (−6.18 + 2.59i)15-s + (−2.00 + 3.46i)16-s + (4.10 − 2.37i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.605 − 0.795i)3-s + (0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.652 − 0.273i)6-s + (−0.298 − 0.954i)7-s + 0.353i·8-s + (−0.267 − 0.963i)9-s + (−0.158 − 0.273i)10-s + (1.29 − 0.750i)11-s + (0.495 + 0.0631i)12-s + 1.00·13-s + (0.154 − 0.689i)14-s + (−0.412 + 0.172i)15-s + (−0.125 + 0.216i)16-s + (0.241 − 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.691 + 0.722i$
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: \((2,\ 210,\ (\ :1),\ 0.691 + 0.722i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.26189 - 0.966066i\)
\(L(\frac12)\) \(\approx\) \(2.26189 - 0.966066i\)
\(L(2)\) 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 \( 1 + (-1.81 + 2.38i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (2.08 + 6.68i)T \)
good11 \( 1 + (-14.2 + 8.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 13.0T + 169T^{2} \)
17 \( 1 + (-4.10 + 2.37i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (16.8 - 29.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (10.0 + 5.82i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 39.6iT - 841T^{2} \)
31 \( 1 + (-26.8 - 46.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (19.9 - 34.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 63.1iT - 1.68e3T^{2} \)
43 \( 1 + 13.9T + 1.84e3T^{2} \)
47 \( 1 + (-9.23 - 5.33i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-76.5 + 44.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (39.2 - 22.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-33.8 + 58.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.4 + 23.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 10.2iT - 5.04e3T^{2} \)
73 \( 1 + (-37.0 - 64.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (41.4 - 71.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 36.0iT - 6.88e3T^{2} \)
89 \( 1 + (52.5 + 30.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 18.6T + 9.40e3T^{2} \)
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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

−12.19051608773783881518484361795, −11.43484821507108716103627379967, −10.03708530816394767009011001953, −8.557502316832561568595208629257, −8.045622131463465129545378016075, −6.68776671494131408719404773965, −6.16597562573225370600030338198, −4.10464243703888072692561323698, −3.41043729321187417026203805374, −1.25236657727735016472693498933, 2.22940839898687709035720274896, 3.56396276178189544134791668216, 4.45159761720252894313537967462, 5.83031208849247428645275935848, 7.04356898374239813592736678289, 8.691135040873223268997660418921, 9.228559566642534257465329610818, 10.42985302257600796320480855692, 11.35708781013583211517401251801, 12.19300987042373157966608520730

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