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Properties

Label 2-210-7.6-c2-0-5
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  − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.23i·5-s − 2.44i·6-s + (1.04 − 6.92i)7-s − 2.82·8-s − 2.99·9-s + 3.16i·10-s − 15.9·11-s + 3.46i·12-s + 6.19i·13-s + (−1.47 + 9.78i)14-s + 3.87·15-s + 4.00·16-s − 19.7i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.447i·5-s − 0.408i·6-s + (0.149 − 0.988i)7-s − 0.353·8-s − 0.333·9-s + 0.316i·10-s − 1.45·11-s + 0.288i·12-s + 0.476i·13-s + (−0.105 + 0.699i)14-s + 0.258·15-s + 0.250·16-s − 1.16i·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: $\chi_{210} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.149 + 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.448755 - 0.521558i\)
\(L(\frac12)\) \(\approx\) \(0.448755 - 0.521558i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - 1.73iT \)
5 \( 1 + 2.23iT \)
7 \( 1 + (-1.04 + 6.92i)T \)
good11 \( 1 + 15.9T + 121T^{2} \)
13 \( 1 - 6.19iT - 169T^{2} \)
17 \( 1 + 19.7iT - 289T^{2} \)
19 \( 1 + 29.3iT - 361T^{2} \)
23 \( 1 - 18.3T + 529T^{2} \)
29 \( 1 + 53.2T + 841T^{2} \)
31 \( 1 + 37.4iT - 961T^{2} \)
37 \( 1 - 61.0T + 1.36e3T^{2} \)
41 \( 1 + 21.3iT - 1.68e3T^{2} \)
43 \( 1 - 25.9T + 1.84e3T^{2} \)
47 \( 1 + 10.3iT - 2.20e3T^{2} \)
53 \( 1 + 64.7T + 2.80e3T^{2} \)
59 \( 1 - 18.0iT - 3.48e3T^{2} \)
61 \( 1 - 91.8iT - 3.72e3T^{2} \)
67 \( 1 + 18.2T + 4.48e3T^{2} \)
71 \( 1 - 108.T + 5.04e3T^{2} \)
73 \( 1 - 73.5iT - 5.32e3T^{2} \)
79 \( 1 + 57.1T + 6.24e3T^{2} \)
83 \( 1 + 36.1iT - 6.88e3T^{2} \)
89 \( 1 - 133. iT - 7.92e3T^{2} \)
97 \( 1 + 29.2iT - 9.40e3T^{2} \)
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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

−11.31968323811860655666700734695, −10.98820912394824709641218248143, −9.752776060306800403956608276745, −9.146755274108111304073712635750, −7.84079147743944960837554542244, −7.11011703945866815742600010262, −5.42134643009890568658092486431, −4.36619473384901134335058506337, −2.65278981412913512114772624132, −0.46378319259128484178661461688, 1.88413778006087952795804223226, 3.13456193556868928046588923907, 5.43228520906035647168329454286, 6.25746729881523433613308034667, 7.74980405067167359695830147721, 8.142946759131314965788695096834, 9.421661716819144543772962870527, 10.54484266669310248636160668971, 11.23413140696649804994310439708, 12.54214865410868521659088316718

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