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Properties

Label 2-210-105.17-c2-0-13
Degree $2$
Conductor $210$
Sign $0.987 - 0.158i$
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  + (1.36 + 0.366i)2-s + (−2.99 − 0.193i)3-s + (1.73 + i)4-s + (4.84 + 1.24i)5-s + (−4.01 − 1.35i)6-s + (−0.400 − 6.98i)7-s + (1.99 + 2i)8-s + (8.92 + 1.15i)9-s + (6.15 + 3.47i)10-s + (−4.21 − 2.43i)11-s + (−4.99 − 3.32i)12-s + (13.3 + 13.3i)13-s + (2.01 − 9.69i)14-s + (−14.2 − 4.67i)15-s + (1.99 + 3.46i)16-s + (25.3 − 6.77i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.997 − 0.0643i)3-s + (0.433 + 0.250i)4-s + (0.968 + 0.249i)5-s + (−0.669 − 0.226i)6-s + (−0.0572 − 0.998i)7-s + (0.249 + 0.250i)8-s + (0.991 + 0.128i)9-s + (0.615 + 0.347i)10-s + (−0.382 − 0.221i)11-s + (−0.416 − 0.277i)12-s + (1.02 + 1.02i)13-s + (0.143 − 0.692i)14-s + (−0.950 − 0.311i)15-s + (0.124 + 0.216i)16-s + (1.48 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.987 - 0.158i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.987 - 0.158i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.05223 + 0.163525i\)
\(L(\frac12)\) \(\approx\) \(2.05223 + 0.163525i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (2.99 + 0.193i)T \)
5 \( 1 + (-4.84 - 1.24i)T \)
7 \( 1 + (0.400 + 6.98i)T \)
good11 \( 1 + (4.21 + 2.43i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-13.3 - 13.3i)T + 169iT^{2} \)
17 \( 1 + (-25.3 + 6.77i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-6.44 - 11.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-5.04 + 18.8i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 41.0T + 841T^{2} \)
31 \( 1 + (-28.4 - 16.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-16.1 + 60.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 50.2T + 1.68e3T^{2} \)
43 \( 1 + (39.5 - 39.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-0.928 + 3.46i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (55.8 - 14.9i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (41.0 + 23.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (58.7 - 33.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-40.3 + 10.7i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 30.7iT - 5.04e3T^{2} \)
73 \( 1 + (-52.4 + 14.0i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (26.6 - 15.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-43.6 + 43.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-54.3 + 31.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-16.7 + 16.7i)T - 9.40e3iT^{2} \)
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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

−12.25167863239141025616611424702, −11.16167213725809663125515260610, −10.49163365283424829921153658340, −9.538293528529372243066218358057, −7.74035272497266843657960295582, −6.69247685537989419978559251341, −5.94050076615244648647059858034, −4.90085933276317213004171463437, −3.54917933744662974148612777666, −1.43593429958416776572700550417, 1.46421076274034714545633197374, 3.23254233562530004909944707166, 5.12488579322873372793010294960, 5.59761864406744940622264482462, 6.44100705988944279908996176869, 8.019122059658594818645922895873, 9.547848428274389188409411162874, 10.27081491186128046290514193435, 11.32205641692176086760004933828, 12.19728618982389908129578992943

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