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Properties

Label 2-210-3.2-c2-0-9
Degree $2$
Conductor $210$
Sign $0.620 + 0.784i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−1.86 − 2.35i)3-s − 2.00·4-s + 2.23i·5-s + (3.32 − 2.63i)6-s + 2.64·7-s − 2.82i·8-s + (−2.07 + 8.75i)9-s − 3.16·10-s − 19.0i·11-s + (3.72 + 4.70i)12-s − 1.55·13-s + 3.74i·14-s + (5.26 − 4.16i)15-s + 4.00·16-s − 28.9i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.620 − 0.784i)3-s − 0.500·4-s + 0.447i·5-s + (0.554 − 0.438i)6-s + 0.377·7-s − 0.353i·8-s + (−0.230 + 0.973i)9-s − 0.316·10-s − 1.73i·11-s + (0.310 + 0.392i)12-s − 0.119·13-s + 0.267i·14-s + (0.350 − 0.277i)15-s + 0.250·16-s − 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.620 + 0.784i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.620 + 0.784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.932430 - 0.451326i\)
\(L(\frac12)\) \(\approx\) \(0.932430 - 0.451326i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (1.86 + 2.35i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 - 2.64T \)
good11 \( 1 + 19.0iT - 121T^{2} \)
13 \( 1 + 1.55T + 169T^{2} \)
17 \( 1 + 28.9iT - 289T^{2} \)
19 \( 1 - 23.3T + 361T^{2} \)
23 \( 1 + 23.7iT - 529T^{2} \)
29 \( 1 + 18.0iT - 841T^{2} \)
31 \( 1 - 7.88T + 961T^{2} \)
37 \( 1 + 20.5T + 1.36e3T^{2} \)
41 \( 1 - 50.8iT - 1.68e3T^{2} \)
43 \( 1 + 31.2T + 1.84e3T^{2} \)
47 \( 1 - 15.7iT - 2.20e3T^{2} \)
53 \( 1 + 78.7iT - 2.80e3T^{2} \)
59 \( 1 - 89.1iT - 3.48e3T^{2} \)
61 \( 1 - 56.4T + 3.72e3T^{2} \)
67 \( 1 - 56.2T + 4.48e3T^{2} \)
71 \( 1 + 28.4iT - 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 + 72.1T + 6.24e3T^{2} \)
83 \( 1 + 104. iT - 6.88e3T^{2} \)
89 \( 1 - 36.3iT - 7.92e3T^{2} \)
97 \( 1 + 57.1T + 9.40e3T^{2} \)
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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

−11.76200142489651385365725450557, −11.36257313044560349935331776035, −10.09479263927927027243867331997, −8.693573369676617789734624688264, −7.76710624031926933324378496423, −6.85405758589820321582395274760, −5.88063907359864452955870269075, −4.94177250645462411399581550992, −2.95718588286098437254588925749, −0.67554090338942628777988336272, 1.61560320777198558239218585429, 3.68460624246411197508965494318, 4.71044333426339228365189124851, 5.59697580120872638807116874615, 7.23934333900540179666817605307, 8.631463198886955434365433577508, 9.700541094167091089486955533946, 10.25626898397338509001470487701, 11.33857707809562504347622628467, 12.20002194599688138013118960204

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