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Properties

Label 2-210-105.17-c2-0-24
Degree $2$
Conductor $210$
Sign $0.737 + 0.674i$
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 + (1.22 − 2.73i)3-s + (1.73 + i)4-s + (4.98 − 0.322i)5-s + (2.67 − 3.29i)6-s + (−4.85 − 5.04i)7-s + (1.99 + 2i)8-s + (−6.00 − 6.70i)9-s + (6.93 + 1.38i)10-s + (17.0 + 9.81i)11-s + (4.85 − 3.52i)12-s + (−7.73 − 7.73i)13-s + (−4.78 − 8.66i)14-s + (5.21 − 14.0i)15-s + (1.99 + 3.46i)16-s + (−12.2 + 3.28i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.407 − 0.913i)3-s + (0.433 + 0.250i)4-s + (0.997 − 0.0645i)5-s + (0.445 − 0.549i)6-s + (−0.693 − 0.720i)7-s + (0.249 + 0.250i)8-s + (−0.667 − 0.744i)9-s + (0.693 + 0.138i)10-s + (1.54 + 0.892i)11-s + (0.404 − 0.293i)12-s + (−0.594 − 0.594i)13-s + (−0.342 − 0.618i)14-s + (0.347 − 0.937i)15-s + (0.124 + 0.216i)16-s + (−0.721 + 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.737 + 0.674i$
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: \((2,\ 210,\ (\ :1),\ 0.737 + 0.674i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.61605 - 1.01591i\)
\(L(\frac12)\) \(\approx\) \(2.61605 - 1.01591i\)
\(L(2)\) 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 \( 1 + (-1.22 + 2.73i)T \)
5 \( 1 + (-4.98 + 0.322i)T \)
7 \( 1 + (4.85 + 5.04i)T \)
good11 \( 1 + (-17.0 - 9.81i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (7.73 + 7.73i)T + 169iT^{2} \)
17 \( 1 + (12.2 - 3.28i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-0.306 - 0.530i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.26 - 12.1i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 29.8T + 841T^{2} \)
31 \( 1 + (17.4 + 10.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (3.54 - 13.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 75.0T + 1.68e3T^{2} \)
43 \( 1 + (46.7 - 46.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (13.4 - 50.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (64.3 - 17.2i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (64.0 + 36.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (19.2 - 11.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-8.82 + 2.36i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 4.20iT - 5.04e3T^{2} \)
73 \( 1 + (-22.9 + 6.15i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (67.0 - 38.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (41.1 - 41.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (-1.42 + 0.825i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-92.5 + 92.5i)T - 9.40e3iT^{2} \)
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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

−12.60354135672027694319661680953, −11.31601559910927443528064062168, −9.891645924035080702399587148491, −9.159584075633365106079434860011, −7.64958493254503103470561251485, −6.68535216519756189306517022230, −6.12727185956894531704106983355, −4.44993357307793977233618793231, −2.97910220604906465112338203616, −1.52981062868684699917779459394, 2.24287236303075968757380752240, 3.41956450950562559126679697394, 4.70665733801281646545524224645, 5.91629428942833795664239113927, 6.69108275526112515496838230044, 8.824712365629326148637490290203, 9.270667120760823477772768078098, 10.25900550324348954457897935465, 11.31887824186604897796000753895, 12.25432163652711645590832852466

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