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Properties

Label 2-210-105.104-c1-0-14
Degree $2$
Conductor $210$
Sign $0.193 + 0.981i$
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  + 2-s + (−0.420 − 1.68i)3-s + 4-s + (−1.95 − 1.08i)5-s + (−0.420 − 1.68i)6-s + (2.37 − 1.16i)7-s + 8-s + (−2.64 + 1.41i)9-s + (−1.95 − 1.08i)10-s − 2.82i·11-s + (−0.420 − 1.68i)12-s + 0.841·13-s + (2.37 − 1.16i)14-s + (−1 + 3.74i)15-s + 16-s − 1.19i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.242 − 0.970i)3-s + 0.5·4-s + (−0.874 − 0.485i)5-s + (−0.171 − 0.685i)6-s + (0.898 − 0.439i)7-s + 0.353·8-s + (−0.881 + 0.471i)9-s + (−0.618 − 0.343i)10-s − 0.852i·11-s + (−0.121 − 0.485i)12-s + 0.233·13-s + (0.635 − 0.311i)14-s + (−0.258 + 0.966i)15-s + 0.250·16-s − 0.288i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.193 + 0.981i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18897 - 0.977839i\)
\(L(\frac12)\) \(\approx\) \(1.18897 - 0.977839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.420 + 1.68i)T \)
5 \( 1 + (1.95 + 1.08i)T \)
7 \( 1 + (-2.37 + 1.16i)T \)
good11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 0.841T + 13T^{2} \)
17 \( 1 + 1.19iT - 17T^{2} \)
19 \( 1 - 4.55iT - 19T^{2} \)
23 \( 1 - 3.29T + 23T^{2} \)
29 \( 1 - 7.98iT - 29T^{2} \)
31 \( 1 - 5.53iT - 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 7.82T + 41T^{2} \)
43 \( 1 - 4.65iT - 43T^{2} \)
47 \( 1 - 4.33iT - 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + 3.91T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 + 4.65iT - 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + 3.06T + 73T^{2} \)
79 \( 1 + 7.29T + 79T^{2} \)
83 \( 1 + 7.70iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 8.11T + 97T^{2} \)
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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

−12.39867879322033635545344819481, −11.18558952126972173640593488302, −10.98462749167802067827556358241, −8.788423240791167026569941747419, −7.905341936243691624450464881653, −7.14021067177172735986427921004, −5.78998376741093680433859475754, −4.75261844088276514191196303240, −3.33607332553714709050354197531, −1.30329009703608405782607126248, 2.73019993213102436039695953845, 4.18505501536016136650475444893, 4.84575213759865007576165003711, 6.17586413167182567630359783380, 7.48469036066271507269149177327, 8.573436771620737537671896778338, 9.862617165545544941529394911738, 11.02753532397347902202182766675, 11.47600495311832576826558908939, 12.31519771906643485921376249849

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