Properties

Degree $2$
Conductor $210$
Sign $0.701 + 0.712i$
Motivic weight $1$
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 + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + 13-s + (−0.499 − 2.59i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-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 + (0.150 + 0.261i)11-s + (0.144 − 0.249i)12-s + 0.277·13-s + (−0.133 − 0.694i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-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: $0.701 + 0.712i$
Motivic weight: \(1\)
Character: $\chi_{210} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49983 - 0.628553i\)
\(L(\frac12)\) \(\approx\) \(1.49983 - 0.628553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-2.5 + 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−12.09952933732796233662109180171, −11.29608300238249074116009719910, −10.35341889859694056514130783466, −9.489981253862455401783347380826, −8.475475833482913151888050460888, −7.26011349959651239983017855611, −5.57402202169901494428638056400, −4.59280485743357610063262082004, −3.51196308069530912728472819133, −1.68821284009841022998340766378, 2.17677101053160040582305696933, 3.79108735080210081194797490015, 5.38337710428824071698166892579, 6.26244529242652534870492501411, 7.44368990681198161912017519393, 8.333648896834752917314253732545, 9.221124652520082017090449300598, 10.66353147211918657398365319474, 11.77263510153502422064628021798, 12.56004300467112020916203357985

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