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Properties

Label 2-210-21.20-c1-0-3
Degree $2$
Conductor $210$
Sign $0.218 - 0.975i$
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  + i·2-s + (1.61 + 0.618i)3-s − 4-s − 5-s + (−0.618 + 1.61i)6-s + (2.61 − 0.381i)7-s − i·8-s + (2.23 + 2.00i)9-s − i·10-s + 4.47i·11-s + (−1.61 − 0.618i)12-s + 1.23i·13-s + (0.381 + 2.61i)14-s + (−1.61 − 0.618i)15-s + 16-s − 5.23·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.934 + 0.356i)3-s − 0.5·4-s − 0.447·5-s + (−0.252 + 0.660i)6-s + (0.989 − 0.144i)7-s − 0.353i·8-s + (0.745 + 0.666i)9-s − 0.316i·10-s + 1.34i·11-s + (−0.467 − 0.178i)12-s + 0.342i·13-s + (0.102 + 0.699i)14-s + (−0.417 − 0.159i)15-s + 0.250·16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18414 + 0.948605i\)
\(L(\frac12)\) \(\approx\) \(1.18414 + 0.948605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.61 - 0.618i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.61 + 0.381i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 - 1.23iT - 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 + 8.47iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 7.70iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 - 0.763T + 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 + 4.94T + 43T^{2} \)
47 \( 1 - 6.47T + 47T^{2} \)
53 \( 1 - 0.472iT - 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 7.23iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 + 0.763iT - 97T^{2} \)
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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

−12.88741595578748830167670406149, −11.55937866114370505271334194582, −10.50101544962496559310016705066, −9.288455303648055474920150732537, −8.583482131391583359230775298980, −7.55090833525974995408141806590, −6.82706276134446434623250687926, −4.68196763603880172432035843732, −4.40572789200858677231059812906, −2.33004092411596175300282581386, 1.57222858037038003848599506127, 3.13685333059573182797924868496, 4.20354530253592862062616267166, 5.79065936957182111881195940933, 7.49038765926742903438116670008, 8.373093804957970469888038664717, 8.944183526097975663103759818318, 10.35834445492418752109551872372, 11.26788830725272053541880992227, 12.11952027186892445508450386445

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