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Properties

Label 2-210-35.12-c1-0-6
Degree $2$
Conductor $210$
Sign $0.874 + 0.484i$
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  + (0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (0.0488 + 2.23i)5-s − i·6-s + (2.15 − 1.52i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.625 + 2.14i)10-s + (0.883 + 1.52i)11-s + (−0.258 − 0.965i)12-s + (−2.71 − 2.71i)13-s + (1.69 − 2.03i)14-s + (2.17 + 0.531i)15-s + (0.500 − 0.866i)16-s + (−2.14 − 0.574i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (0.0218 + 0.999i)5-s − 0.408i·6-s + (0.816 − 0.577i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.197 + 0.678i)10-s + (0.266 + 0.461i)11-s + (−0.0747 − 0.278i)12-s + (−0.752 − 0.752i)13-s + (0.451 − 0.543i)14-s + (0.560 + 0.137i)15-s + (0.125 − 0.216i)16-s + (−0.520 − 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.874 + 0.484i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81108 - 0.468386i\)
\(L(\frac12)\) \(\approx\) \(1.81108 - 0.468386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-0.0488 - 2.23i)T \)
7 \( 1 + (-2.15 + 1.52i)T \)
good11 \( 1 + (-0.883 - 1.52i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.71 + 2.71i)T + 13iT^{2} \)
17 \( 1 + (2.14 + 0.574i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.886 - 1.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.04 - 3.90i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.84iT - 29T^{2} \)
31 \( 1 + (8.94 - 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.21 + 0.861i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + (-3.46 + 3.46i)T - 43iT^{2} \)
47 \( 1 + (-1.59 - 5.93i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.396 + 0.106i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.18 + 8.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.87 + 3.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.97 + 7.37i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (-2.75 + 10.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-10.9 - 6.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.94 - 1.94i)T + 83iT^{2} \)
89 \( 1 + (-0.558 + 0.966i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.26 + 7.26i)T - 97iT^{2} \)
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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

−12.35639365605126852193711134167, −11.33929156137757496326147651233, −10.68437126010033160343108504725, −9.554421276111047742127351152777, −7.81162435982866809316868394877, −7.25566746519763511907457463860, −6.11675799689493144735559247862, −4.75856776406870704249624774025, −3.34987986563469409282303515094, −1.95645094532689080246629341073, 2.19954396857977002512350157580, 4.09180941992144372457299315812, 4.90921126222308171058953267695, 5.85904254067007186761191375456, 7.40153988890058985930038816538, 8.696999854054805380089561997407, 9.166170226183036627794805977117, 10.74057468763794271605483942995, 11.65429123266432618587859802850, 12.43037099171580270182556039370

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