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Properties

Label 2-210-105.38-c2-0-14
Degree $2$
Conductor $210$
Sign $0.618 - 0.785i$
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  + (−0.366 + 1.36i)2-s + (2.66 − 1.37i)3-s + (−1.73 − i)4-s + (−3.20 + 3.83i)5-s + (0.898 + 4.14i)6-s + (6.79 − 1.68i)7-s + (2 − 1.99i)8-s + (5.23 − 7.32i)9-s + (−4.06 − 5.78i)10-s + (13.7 + 7.94i)11-s + (−5.99 − 0.290i)12-s + (−5.32 + 5.32i)13-s + (−0.185 + 9.89i)14-s + (−3.29 + 14.6i)15-s + (1.99 + 3.46i)16-s + (4.55 + 16.9i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.889 − 0.457i)3-s + (−0.433 − 0.250i)4-s + (−0.641 + 0.767i)5-s + (0.149 + 0.691i)6-s + (0.970 − 0.240i)7-s + (0.250 − 0.249i)8-s + (0.581 − 0.813i)9-s + (−0.406 − 0.578i)10-s + (1.25 + 0.722i)11-s + (−0.499 − 0.0242i)12-s + (−0.409 + 0.409i)13-s + (−0.0132 + 0.706i)14-s + (−0.219 + 0.975i)15-s + (0.124 + 0.216i)16-s + (0.267 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.618 - 0.785i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.67243 + 0.811854i\)
\(L(\frac12)\) \(\approx\) \(1.67243 + 0.811854i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + (-2.66 + 1.37i)T \)
5 \( 1 + (3.20 - 3.83i)T \)
7 \( 1 + (-6.79 + 1.68i)T \)
good11 \( 1 + (-13.7 - 7.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.32 - 5.32i)T - 169iT^{2} \)
17 \( 1 + (-4.55 - 16.9i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (5.62 + 9.74i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-39.1 - 10.4i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 22.8T + 841T^{2} \)
31 \( 1 + (-36.0 - 20.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (28.8 + 7.72i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 55.0T + 1.68e3T^{2} \)
43 \( 1 + (14.2 + 14.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (25.2 + 6.77i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (15.8 + 59.0i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-10.7 - 6.18i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-14.5 + 8.40i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (27.8 + 103. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 34.7iT - 5.04e3T^{2} \)
73 \( 1 + (15.2 + 56.8i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (40.5 - 23.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (75.6 + 75.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (116. - 67.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-75.7 - 75.7i)T + 9.40e3iT^{2} \)
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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

−12.30035359286730312875641734338, −11.38351095723199165730752967938, −10.15336806572084222673105259398, −8.987289340192808506654887574986, −8.181170005641878972176149377905, −7.12748882576387263920920336161, −6.71490832175527623025749933354, −4.68859649196308116468531163169, −3.56033138419937876901104503035, −1.64871943781210428811078340070, 1.29130919147578098427972716541, 3.04523485663267272321604302358, 4.25796543026745948109591254607, 5.15450432353875463326934715725, 7.36318655538365359062404228306, 8.450782829146424456575089051925, 8.872467486928959157649672612828, 9.916419388452118656659615573996, 11.20996462746719870874809054304, 11.79827972018468622500136269453

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