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Properties

Label 2-210-105.38-c2-0-22
Degree $2$
Conductor $210$
Sign $-0.656 + 0.754i$
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 + (−1.73 − 2.44i)3-s + (−1.73 − i)4-s + (−0.120 + 4.99i)5-s + (3.97 − 1.47i)6-s + (4.07 − 5.69i)7-s + (2 − 1.99i)8-s + (−2.97 + 8.49i)9-s + (−6.78 − 1.99i)10-s + (−12.1 − 7.01i)11-s + (0.557 + 5.97i)12-s + (−7.32 + 7.32i)13-s + (6.28 + 7.64i)14-s + (12.4 − 8.37i)15-s + (1.99 + 3.46i)16-s + (−2.35 − 8.77i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.578 − 0.815i)3-s + (−0.433 − 0.250i)4-s + (−0.0241 + 0.999i)5-s + (0.663 − 0.245i)6-s + (0.581 − 0.813i)7-s + (0.250 − 0.249i)8-s + (−0.331 + 0.943i)9-s + (−0.678 − 0.199i)10-s + (−1.10 − 0.637i)11-s + (0.0464 + 0.497i)12-s + (−0.563 + 0.563i)13-s + (0.449 + 0.546i)14-s + (0.829 − 0.558i)15-s + (0.124 + 0.216i)16-s + (−0.138 − 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.656 + 0.754i$
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: \((2,\ 210,\ (\ :1),\ -0.656 + 0.754i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.118098 - 0.259501i\)
\(L(\frac12)\) \(\approx\) \(0.118098 - 0.259501i\)
\(L(2)\) 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 \( 1 + (1.73 + 2.44i)T \)
5 \( 1 + (0.120 - 4.99i)T \)
7 \( 1 + (-4.07 + 5.69i)T \)
good11 \( 1 + (12.1 + 7.01i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (7.32 - 7.32i)T - 169iT^{2} \)
17 \( 1 + (2.35 + 8.77i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (13.9 + 24.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (31.8 + 8.52i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 4.58T + 841T^{2} \)
31 \( 1 + (31.4 + 18.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-36.6 - 9.81i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 53.2T + 1.68e3T^{2} \)
43 \( 1 + (-16.6 - 16.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-17.2 - 4.62i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-21.1 - 78.9i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (8.07 + 4.66i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-92.7 + 53.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-24.2 - 90.5i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 31.9iT - 5.04e3T^{2} \)
73 \( 1 + (17.1 + 63.8i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (46.9 - 27.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (37.9 + 37.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (91.9 - 53.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (59.6 + 59.6i)T + 9.40e3iT^{2} \)
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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

−11.54029269883241326130929286934, −10.91713793874092634677261806548, −10.02280211121766923199931389716, −8.352578147964228419124910479149, −7.47483076852712903482015810402, −6.84818546771938967993006376385, −5.76019181413062252994015102920, −4.49270952686409044387817204444, −2.39174013258946082785873330070, −0.17169386966528105870461977246, 2.03543719341453190021100496085, 3.94233385772039233458106418503, 5.05678122482433285221197954150, 5.72429618979808741118104857931, 7.925915379424504958504079858529, 8.666604489342874412106698233863, 9.844951033458439444427833234620, 10.39972192148103925102171253110, 11.60556479707541308444053699040, 12.37317748878686038129153576368

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