Properties

Label 2-210-35.34-c2-0-9
Degree $2$
Conductor $210$
Sign $0.896 + 0.443i$
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  − 1.41i·2-s + 1.73·3-s − 2.00·4-s + (4.91 + 0.905i)5-s − 2.44i·6-s + (−1.91 + 6.73i)7-s + 2.82i·8-s + 2.99·9-s + (1.28 − 6.95i)10-s + 17.5·11-s − 3.46·12-s + 4.83·13-s + (9.52 + 2.70i)14-s + (8.51 + 1.56i)15-s + 4.00·16-s − 18.0·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.500·4-s + (0.983 + 0.181i)5-s − 0.408i·6-s + (−0.273 + 0.961i)7-s + 0.353i·8-s + 0.333·9-s + (0.128 − 0.695i)10-s + 1.59·11-s − 0.288·12-s + 0.371·13-s + (0.680 + 0.193i)14-s + (0.567 + 0.104i)15-s + 0.250·16-s − 1.06·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.896 + 0.443i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.04501 - 0.477812i\)
\(L(\frac12)\) \(\approx\) \(2.04501 - 0.477812i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 - 1.73T \)
5 \( 1 + (-4.91 - 0.905i)T \)
7 \( 1 + (1.91 - 6.73i)T \)
good11 \( 1 - 17.5T + 121T^{2} \)
13 \( 1 - 4.83T + 169T^{2} \)
17 \( 1 + 18.0T + 289T^{2} \)
19 \( 1 + 9.13iT - 361T^{2} \)
23 \( 1 - 3.72iT - 529T^{2} \)
29 \( 1 - 1.12T + 841T^{2} \)
31 \( 1 + 57.0iT - 961T^{2} \)
37 \( 1 - 41.3iT - 1.36e3T^{2} \)
41 \( 1 + 11.7iT - 1.68e3T^{2} \)
43 \( 1 + 64.4iT - 1.84e3T^{2} \)
47 \( 1 + 77.6T + 2.20e3T^{2} \)
53 \( 1 - 77.5iT - 2.80e3T^{2} \)
59 \( 1 - 87.0iT - 3.48e3T^{2} \)
61 \( 1 + 5.36iT - 3.72e3T^{2} \)
67 \( 1 - 47.1iT - 4.48e3T^{2} \)
71 \( 1 + 58.3T + 5.04e3T^{2} \)
73 \( 1 - 53.4T + 5.32e3T^{2} \)
79 \( 1 + 74.9T + 6.24e3T^{2} \)
83 \( 1 + 28.7T + 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 + 107.T + 9.40e3T^{2} \)
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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

−12.01314862402458521201303171168, −11.19645136427877266446223647455, −9.900855323944730046809182808522, −9.181494387920937154583195443671, −8.630837082846022670270907246519, −6.79206804871290756569213554464, −5.83126919638409598579459826700, −4.26051181051663124034320391566, −2.84085022297549423779423461073, −1.71394249144172783993118118325, 1.46099022246911266346196111961, 3.56866686994097369935731044932, 4.73014821527112965276138390563, 6.37443085586517271041296174392, 6.84183854748917786164176520211, 8.314179148127887958486409638002, 9.200270344149919236824480203421, 9.909739282115624395688051851748, 11.07234294516361555248564811461, 12.63213942684833597388141478357

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