Properties

Label 2-210-5.4-c1-0-0
Degree $2$
Conductor $210$
Sign $-0.894 - 0.447i$
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 + i·3-s − 4-s + (−1 + 2i)5-s − 6-s + i·7-s i·8-s − 9-s + (−2 − i)10-s − 2·11-s i·12-s − 2i·13-s − 14-s + (−2 − i)15-s + 16-s + 8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.408·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.603·11-s − 0.288i·12-s − 0.554i·13-s − 0.267·14-s + (−0.516 − 0.258i)15-s + 0.250·16-s + 1.94i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.214715 + 0.909549i\)
\(L(\frac12)\) \(\approx\) \(0.214715 + 0.909549i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (1 - 2i)T \)
7 \( 1 - iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−12.85341074076681028669562334064, −11.77637777828850373241937052240, −10.55799777997487066299444878398, −10.05410291922253338571499584509, −8.534746866712033934298779236000, −7.896812855572944601763094997567, −6.56577000271706956530288027022, −5.62030931674211299426613422821, −4.24627869961818121660718616196, −2.96870758936338633814540348254, 0.841117477816508379795875374473, 2.70745549179855071060634531176, 4.34401055138544843684914808244, 5.34915356396605090891664883229, 7.05683691550879363480497476428, 8.015255240138304758518364371083, 9.049255686670733671635867707160, 9.976663410368504927093723199896, 11.36285221408295252654928188196, 11.87516824747386644714238969213

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