Properties

Label 2-210-105.2-c1-0-8
Degree $2$
Conductor $210$
Sign $0.969 + 0.243i$
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  + (0.258 + 0.965i)2-s + (−1.65 − 0.517i)3-s + (−0.866 + 0.499i)4-s + (−0.593 − 2.15i)5-s + (0.0717 − 1.73i)6-s + (2.64 + 0.113i)7-s + (−0.707 − 0.707i)8-s + (2.46 + 1.70i)9-s + (1.92 − 1.13i)10-s + (3.54 − 2.04i)11-s + (1.69 − 0.378i)12-s + (3.69 − 3.69i)13-s + (0.574 + 2.58i)14-s + (−0.134 + 3.87i)15-s + (0.500 − 0.866i)16-s + (−7.20 − 1.93i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.954 − 0.298i)3-s + (−0.433 + 0.249i)4-s + (−0.265 − 0.964i)5-s + (0.0292 − 0.706i)6-s + (0.999 + 0.0427i)7-s + (−0.249 − 0.249i)8-s + (0.821 + 0.569i)9-s + (0.609 − 0.357i)10-s + (1.06 − 0.616i)11-s + (0.487 − 0.109i)12-s + (1.02 − 1.02i)13-s + (0.153 + 0.690i)14-s + (−0.0346 + 0.999i)15-s + (0.125 − 0.216i)16-s + (−1.74 − 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.969 + 0.243i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01718 - 0.125670i\)
\(L(\frac12)\) \(\approx\) \(1.01718 - 0.125670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (1.65 + 0.517i)T \)
5 \( 1 + (0.593 + 2.15i)T \)
7 \( 1 + (-2.64 - 0.113i)T \)
good11 \( 1 + (-3.54 + 2.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.69 + 3.69i)T - 13iT^{2} \)
17 \( 1 + (7.20 + 1.93i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.12 - 1.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.06 + 0.284i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 + (-2.62 - 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.34 - 1.43i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.74iT - 41T^{2} \)
43 \( 1 + (-1.27 + 1.27i)T - 43iT^{2} \)
47 \( 1 + (-0.796 - 2.97i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.47 - 5.50i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.09 - 8.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.814 - 1.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.909 + 3.39i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.89iT - 71T^{2} \)
73 \( 1 + (5.38 + 1.44i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.18 + 4.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.31 - 6.31i)T + 83iT^{2} \)
89 \( 1 + (6.71 - 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.21 + 5.21i)T + 97iT^{2} \)
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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

−12.28038050514105714854102954680, −11.53386793956054080025188425649, −10.67699906937542685416564831311, −8.956659519779041097514968034049, −8.338976610324735778117650792072, −7.13878186425222785284278632761, −5.98333002249326637903900407706, −5.08578621289871711436449785996, −4.10348676062208825695244609929, −1.11284956439841049387029181208, 1.78063715512349444359264331861, 3.91441683616568318175472317060, 4.61517842930768518097415542321, 6.24528818875189172884288534304, 7.00122050617295476738578577235, 8.678999841596617652208817923136, 9.767712799678581196540470527762, 10.93197464741777831961518939412, 11.35594414908406829995752329086, 11.89131330887571656518643694992

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