Properties

Label 2-210-35.3-c1-0-4
Degree $2$
Conductor $210$
Sign $0.980 - 0.198i$
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 + (1.87 + 1.22i)5-s i·6-s + (−0.942 + 2.47i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (1.48 + 1.66i)10-s + (1.55 − 2.69i)11-s + (0.258 − 0.965i)12-s + (3.40 − 3.40i)13-s + (−1.55 + 2.14i)14-s + (0.700 − 2.12i)15-s + (0.500 + 0.866i)16-s + (−5.14 + 1.37i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (0.836 + 0.548i)5-s − 0.408i·6-s + (−0.356 + 0.934i)7-s + (0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.470 + 0.527i)10-s + (0.468 − 0.811i)11-s + (0.0747 − 0.278i)12-s + (0.945 − 0.945i)13-s + (−0.414 + 0.572i)14-s + (0.180 − 0.548i)15-s + (0.125 + 0.216i)16-s + (−1.24 + 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80036 + 0.180047i\)
\(L(\frac12)\) \(\approx\) \(1.80036 + 0.180047i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 - 1.22i)T \)
7 \( 1 + (0.942 - 2.47i)T \)
good11 \( 1 + (-1.55 + 2.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.40 + 3.40i)T - 13iT^{2} \)
17 \( 1 + (5.14 - 1.37i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.61 + 6.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.36 - 5.08i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.49iT - 29T^{2} \)
31 \( 1 + (7.98 + 4.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.47 - 0.929i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (3.86 + 3.86i)T + 43iT^{2} \)
47 \( 1 + (1.05 - 3.94i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.32 - 0.890i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.666 + 1.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.8 + 6.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.72 + 6.43i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (-2.13 - 7.95i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.65 + 0.957i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.97 - 8.97i)T - 83iT^{2} \)
89 \( 1 + (-2.03 - 3.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.69 + 2.69i)T + 97iT^{2} \)
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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

−12.88043172101633548672273660708, −11.30358001129692442195294409938, −10.95345649374273731323857389718, −9.296565576179036685476529510190, −8.426631793596665948894281709402, −6.86553834183219368291616198862, −6.14726682685246447933334096061, −5.39903259862903356496216619563, −3.42577753222897876242330722361, −2.17881709526366201221656885692, 1.87854626798839844852588571742, 3.95094810481860139312312722613, 4.58094652651890539319845610924, 6.09102181015340700348264081524, 6.79508106562984955708502854349, 8.590013377246379738973534882800, 9.638879627446065561316248578476, 10.40752809894505416018298124489, 11.34608042968256416311107535663, 12.53385556313873897409064455789

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