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Properties

Label 2-210-105.83-c2-0-21
Degree $2$
Conductor $210$
Sign $0.318 + 0.947i$
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 − i)2-s + (−1.21 + 2.74i)3-s + 2i·4-s + (3.32 − 3.73i)5-s + (3.95 − 1.52i)6-s + (−6.61 − 2.29i)7-s + (2 − 2i)8-s + (−6.03 − 6.67i)9-s + (−7.05 + 0.417i)10-s + 10.5i·11-s + (−5.48 − 2.43i)12-s + (14.9 − 14.9i)13-s + (4.31 + 8.90i)14-s + (6.20 + 13.6i)15-s − 4·16-s + (15.4 − 15.4i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.405 + 0.913i)3-s + 0.5i·4-s + (0.664 − 0.747i)5-s + (0.659 − 0.254i)6-s + (−0.944 − 0.327i)7-s + (0.250 − 0.250i)8-s + (−0.670 − 0.741i)9-s + (−0.705 + 0.0417i)10-s + 0.961i·11-s + (−0.456 − 0.202i)12-s + (1.15 − 1.15i)13-s + (0.308 + 0.636i)14-s + (0.413 + 0.910i)15-s − 0.250·16-s + (0.911 − 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.318 + 0.947i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.764369 - 0.549451i\)
\(L(\frac12)\) \(\approx\) \(0.764369 - 0.549451i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (1.21 - 2.74i)T \)
5 \( 1 + (-3.32 + 3.73i)T \)
7 \( 1 + (6.61 + 2.29i)T \)
good11 \( 1 - 10.5iT - 121T^{2} \)
13 \( 1 + (-14.9 + 14.9i)T - 169iT^{2} \)
17 \( 1 + (-15.4 + 15.4i)T - 289iT^{2} \)
19 \( 1 + 17.3T + 361T^{2} \)
23 \( 1 + (-23.1 + 23.1i)T - 529iT^{2} \)
29 \( 1 - 23.7T + 841T^{2} \)
31 \( 1 + 33.1iT - 961T^{2} \)
37 \( 1 + (17.6 - 17.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 11.8T + 1.68e3T^{2} \)
43 \( 1 + (22.8 + 22.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-12.6 + 12.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-15.3 + 15.3i)T - 2.80e3iT^{2} \)
59 \( 1 + 31.0iT - 3.48e3T^{2} \)
61 \( 1 - 48.6iT - 3.72e3T^{2} \)
67 \( 1 + (77.5 - 77.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 60.7iT - 5.04e3T^{2} \)
73 \( 1 + (-3.52 + 3.52i)T - 5.32e3iT^{2} \)
79 \( 1 + 99.4iT - 6.24e3T^{2} \)
83 \( 1 + (-16.9 - 16.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 17.6iT - 7.92e3T^{2} \)
97 \( 1 + (34.7 + 34.7i)T + 9.40e3iT^{2} \)
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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

−11.93466999138225339543732260064, −10.52097191494083836875333483543, −10.14371326425731602887942261600, −9.253776233425263064813582004472, −8.380265879603198548046084664969, −6.70077467816957727604633644891, −5.52087569782235359284714978957, −4.30288271670881539228606396571, −2.94556192807030361833316852945, −0.70474089958181528098046882597, 1.51484764938383600636000144372, 3.22728474617021400726303282380, 5.65302001657038641928334117733, 6.32477967342986768290092643886, 6.92758982250962424875778811112, 8.353787601306633765718464388133, 9.190923926367424097201891675731, 10.50361513602889776328515771071, 11.15729311691778755927410702930, 12.38688357076596517720553745156

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