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Properties

Label 2-210-15.14-c2-0-3
Degree $2$
Conductor $210$
Sign $0.833 - 0.552i$
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 + (−1.70 − 2.46i)3-s + 2.00·4-s + (−4.99 − 0.104i)5-s + (2.41 + 3.48i)6-s + 2.64i·7-s − 2.82·8-s + (−3.15 + 8.42i)9-s + (7.06 + 0.148i)10-s − 2.66i·11-s + (−3.41 − 4.93i)12-s + 5.56i·13-s − 3.74i·14-s + (8.28 + 12.5i)15-s + 4.00·16-s + 18.5·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.569 − 0.821i)3-s + 0.500·4-s + (−0.999 − 0.0209i)5-s + (0.402 + 0.581i)6-s + 0.377i·7-s − 0.353·8-s + (−0.350 + 0.936i)9-s + (0.706 + 0.0148i)10-s − 0.242i·11-s + (−0.284 − 0.410i)12-s + 0.427i·13-s − 0.267i·14-s + (0.552 + 0.833i)15-s + 0.250·16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.833 - 0.552i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.833 - 0.552i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.587719 + 0.177044i\)
\(L(\frac12)\) \(\approx\) \(0.587719 + 0.177044i\)
\(L(2)\) 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 \( 1 + (1.70 + 2.46i)T \)
5 \( 1 + (4.99 + 0.104i)T \)
7 \( 1 - 2.64iT \)
good11 \( 1 + 2.66iT - 121T^{2} \)
13 \( 1 - 5.56iT - 169T^{2} \)
17 \( 1 - 18.5T + 289T^{2} \)
19 \( 1 - 0.634T + 361T^{2} \)
23 \( 1 - 15.6T + 529T^{2} \)
29 \( 1 - 43.3iT - 841T^{2} \)
31 \( 1 - 13.5T + 961T^{2} \)
37 \( 1 - 35.0iT - 1.36e3T^{2} \)
41 \( 1 + 15.6iT - 1.68e3T^{2} \)
43 \( 1 - 64.4iT - 1.84e3T^{2} \)
47 \( 1 - 52.5T + 2.20e3T^{2} \)
53 \( 1 + 51.6T + 2.80e3T^{2} \)
59 \( 1 + 32.9iT - 3.48e3T^{2} \)
61 \( 1 + 104.T + 3.72e3T^{2} \)
67 \( 1 - 113. iT - 4.48e3T^{2} \)
71 \( 1 - 36.8iT - 5.04e3T^{2} \)
73 \( 1 + 144. iT - 5.32e3T^{2} \)
79 \( 1 - 57.9T + 6.24e3T^{2} \)
83 \( 1 + 45.0T + 6.88e3T^{2} \)
89 \( 1 - 80.9iT - 7.92e3T^{2} \)
97 \( 1 - 96.2iT - 9.40e3T^{2} \)
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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

−12.07741129653224369768634645929, −11.35219063692552703831962756286, −10.51241370197886281432866479882, −9.061880721963161992425465671907, −8.092524115460126126082633936440, −7.32026955612368627638896781496, −6.30839645738414275583956955714, −4.97808983534697921972536106745, −3.06372807612780910687904014675, −1.17129988950980612097811650960, 0.56503209928903547580990749724, 3.25719875591171410319914739501, 4.43648415932437805751476444579, 5.79187499707413178652163188329, 7.15382013978260910948429937120, 8.064697134159333035193712665956, 9.223959599452274059280467481681, 10.20772217480265728018985282688, 10.92431062505325559154682692430, 11.81885989203791635252910477498

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