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Properties

Label 2-210-7.4-c1-0-3
Degree $2$
Conductor $210$
Sign $-0.701 + 0.712i$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + (−2 − 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (2.5 − 4.33i)11-s + (0.499 + 0.866i)12-s − 5·13-s + (−0.499 + 2.59i)14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.408·6-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.753 − 1.30i)11-s + (0.144 + 0.249i)12-s − 1.38·13-s + (−0.133 + 0.694i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342406 - 0.817037i\)
\(L(\frac12)\) \(\approx\) \(0.342406 - 0.817037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2 + 1.73i)T \)
good11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (-5.5 - 9.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−12.14309058434703106274816141330, −11.14424201069254828347079896328, −9.864281256627199700013985873582, −9.249933012113699635041113890814, −7.993585237523661689107967609075, −7.20836337470889214761053674365, −5.78353454051427504962569689774, −4.02662175875836488805177239606, −2.90214945622279744677650796997, −0.856922317882564226388418665597, 2.58867224981759117577008902522, 4.22722423404120236916766006798, 5.46957335753293207276971995174, 6.83736112334465588892998076990, 7.55210403002109270418447210492, 9.031514446498088448474668699270, 9.588859492014329857682023800138, 10.41788120155298777952940914741, 11.86170958269909143406564398218, 12.62648675376300103373547009149

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