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Properties

Label 2-210-21.2-c2-0-17
Degree $2$
Conductor $210$
Sign $-0.149 + 0.988i$
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  + (−1.22 − 0.707i)2-s + (1.03 − 2.81i)3-s + (0.999 + 1.73i)4-s + (1.93 + 1.11i)5-s + (−3.25 + 2.71i)6-s + (5.52 − 4.30i)7-s − 2.82i·8-s + (−6.86 − 5.81i)9-s + (−1.58 − 2.73i)10-s + (−1.44 + 0.836i)11-s + (5.91 − 1.02i)12-s + 20.4·13-s + (−9.80 + 1.36i)14-s + (5.14 − 4.29i)15-s + (−2.00 + 3.46i)16-s + (−3.30 + 1.90i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.344 − 0.938i)3-s + (0.249 + 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.542 + 0.453i)6-s + (0.788 − 0.614i)7-s − 0.353i·8-s + (−0.762 − 0.646i)9-s + (−0.158 − 0.273i)10-s + (−0.131 + 0.0760i)11-s + (0.492 − 0.0856i)12-s + 1.57·13-s + (−0.700 + 0.0973i)14-s + (0.343 − 0.286i)15-s + (−0.125 + 0.216i)16-s + (−0.194 + 0.112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.149 + 0.988i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.149 + 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.939180 - 1.09168i\)
\(L(\frac12)\) \(\approx\) \(0.939180 - 1.09168i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-1.03 + 2.81i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (-5.52 + 4.30i)T \)
good11 \( 1 + (1.44 - 0.836i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 20.4T + 169T^{2} \)
17 \( 1 + (3.30 - 1.90i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.08 + 14.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (26.8 + 15.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 15.5iT - 841T^{2} \)
31 \( 1 + (21.5 + 37.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (15.1 - 26.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 45.4iT - 1.68e3T^{2} \)
43 \( 1 - 33.1T + 1.84e3T^{2} \)
47 \( 1 + (-18.0 - 10.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-29.2 + 16.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-67.5 + 38.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (44.2 - 76.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.6 - 30.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + (-65.5 - 113. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-26.0 + 45.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 72.2iT - 6.88e3T^{2} \)
89 \( 1 + (79.2 + 45.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 34.3T + 9.40e3T^{2} \)
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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

−11.61238751400375742281715702401, −11.07402322343715472713280146315, −9.918977519536942638993047224425, −8.685376569614549564256720099153, −8.006934385219692537047481455675, −6.97315930790724617331085276387, −5.91103999891566626654426277057, −3.93313637622994458557782718739, −2.34354758031558440677467342382, −1.05451559355056845296453306354, 1.84255290557954606292893237711, 3.68023092801929204861379370413, 5.23235639598551149298671749011, 5.96937656252559292058952663289, 7.74099235605783920344430784840, 8.674541907033034868281221543215, 9.190800342198749260366781846184, 10.42784064600078358565581909785, 11.05744082298554435479920826556, 12.19588674359506523567711134908

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