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Properties

Label 2-210-35.4-c1-0-4
Degree $2$
Conductor $210$
Sign $0.999 + 0.00412i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−1.40 + 1.74i)5-s + 0.999·6-s + (2.51 + 0.806i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.344 + 2.20i)10-s + (2.39 − 4.14i)11-s + (0.866 − 0.499i)12-s + 3.17i·13-s + (2.58 − 0.561i)14-s + (−2.08 + 0.806i)15-s + (−0.5 − 0.866i)16-s + (−4.52 − 2.61i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.627 + 0.778i)5-s + 0.408·6-s + (0.952 + 0.304i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.108 + 0.698i)10-s + (0.721 − 1.24i)11-s + (0.249 − 0.144i)12-s + 0.879i·13-s + (0.691 − 0.150i)14-s + (−0.538 + 0.208i)15-s + (−0.125 − 0.216i)16-s + (−1.09 − 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.999 + 0.00412i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.999 + 0.00412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86279 - 0.00384614i\)
\(L(\frac12)\) \(\approx\) \(1.86279 - 0.00384614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.40 - 1.74i)T \)
7 \( 1 + (-2.51 - 0.806i)T \)
good11 \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.17iT - 13T^{2} \)
17 \( 1 + (4.52 + 2.61i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.58 + 2.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.21 - 3.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 + (2.08 - 3.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.21 + 3.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.05T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-9.97 + 5.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.02 + 4.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.36 - 9.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.55 + 6.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.28 + 4.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.08 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.39iT - 83T^{2} \)
89 \( 1 + (-4.78 - 8.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.27iT - 97T^{2} \)
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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

−12.12082421852596853001762184593, −11.25793451842435427820500876404, −10.96496586666614498732694749128, −9.377394120960814994590564409264, −8.476260916646274556771374148427, −7.25629694209065545484842718082, −6.08981844504480088910461612731, −4.55545557001232425579490920050, −3.63262841114390890811497598370, −2.23008101522047363477640887926, 1.90260332856618686789969354837, 4.01173288424606710182633207537, 4.60925263765311503814384853582, 6.15699105848992261103530320961, 7.55032379730389619356343724990, 8.071192513024162168670154516520, 9.120103221756621446152879563053, 10.56237300770255017828927738752, 11.78200676369638946214251744298, 12.49889415297053287888749894311

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