Properties

Label 2-210-105.68-c2-0-6
Degree $2$
Conductor $210$
Sign $-0.923 - 0.384i$
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 + (0.941 + 2.84i)3-s + (1.73 − i)4-s + (2.72 + 4.19i)5-s + (−2.32 − 3.54i)6-s + (−5.35 + 4.50i)7-s + (−1.99 + 2i)8-s + (−7.22 + 5.36i)9-s + (−5.25 − 4.73i)10-s + (−8.15 + 4.70i)11-s + (4.47 + 3.99i)12-s + (16.1 − 16.1i)13-s + (5.66 − 8.11i)14-s + (−9.37 + 11.7i)15-s + (1.99 − 3.46i)16-s + (9.03 + 2.42i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.313 + 0.949i)3-s + (0.433 − 0.250i)4-s + (0.544 + 0.838i)5-s + (−0.388 − 0.591i)6-s + (−0.765 + 0.643i)7-s + (−0.249 + 0.250i)8-s + (−0.802 + 0.596i)9-s + (−0.525 − 0.473i)10-s + (−0.741 + 0.428i)11-s + (0.373 + 0.332i)12-s + (1.24 − 1.24i)13-s + (0.404 − 0.579i)14-s + (−0.625 + 0.780i)15-s + (0.124 − 0.216i)16-s + (0.531 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.923 - 0.384i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.923 - 0.384i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.195457 + 0.977742i\)
\(L(\frac12)\) \(\approx\) \(0.195457 + 0.977742i\)
\(L(2)\) 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 \( 1 + (-0.941 - 2.84i)T \)
5 \( 1 + (-2.72 - 4.19i)T \)
7 \( 1 + (5.35 - 4.50i)T \)
good11 \( 1 + (8.15 - 4.70i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-16.1 + 16.1i)T - 169iT^{2} \)
17 \( 1 + (-9.03 - 2.42i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (13.1 - 22.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (6.16 + 23.0i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 34.0T + 841T^{2} \)
31 \( 1 + (-2.88 + 1.66i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-12.3 - 46.2i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 3.31T + 1.68e3T^{2} \)
43 \( 1 + (-28.9 - 28.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (0.830 + 3.09i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-56.7 - 15.2i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (33.5 - 19.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-80.5 - 46.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-3.66 - 0.981i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 26.4iT - 5.04e3T^{2} \)
73 \( 1 + (-27.3 - 7.31i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-38.5 - 22.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-70.6 - 70.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-118. - 68.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (118. + 118. i)T + 9.40e3iT^{2} \)
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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

−12.58211145086673036931095430991, −11.08423953053104800757913748283, −10.27760277715211181034766817784, −9.895611092611327135120486823525, −8.673576779846331603217794568909, −7.83899869446597024615073571629, −6.21234725135039440698773888767, −5.58230804382563044656531421432, −3.53986317184568405502947946633, −2.46235333114679471334553395141, 0.65175778701799644335082374552, 2.07032137862268001702652954576, 3.71467137356730193521489017424, 5.74564343146975406093934841676, 6.72900017778759086321925035660, 7.76890349725558925880566380653, 8.894990425538976108695195622127, 9.386021832062671068772088846944, 10.76300035449772601062528626858, 11.69565928157137964891422817945

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