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Properties

Label 2-210-105.83-c2-0-1
Degree $2$
Conductor $210$
Sign $-0.905 + 0.424i$
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 − i)2-s + (−0.282 + 2.98i)3-s + 2i·4-s + (−3.28 + 3.77i)5-s + (3.26 − 2.70i)6-s + (−3.67 + 5.95i)7-s + (2 − 2i)8-s + (−8.84 − 1.68i)9-s + (7.05 − 0.487i)10-s − 19.5i·11-s + (−5.97 − 0.565i)12-s + (−2.90 + 2.90i)13-s + (9.63 − 2.28i)14-s + (−10.3 − 10.8i)15-s − 4·16-s + (16.3 − 16.3i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.0942 + 0.995i)3-s + 0.5i·4-s + (−0.656 + 0.754i)5-s + (0.544 − 0.450i)6-s + (−0.525 + 0.850i)7-s + (0.250 − 0.250i)8-s + (−0.982 − 0.187i)9-s + (0.705 − 0.0487i)10-s − 1.77i·11-s + (−0.497 − 0.0471i)12-s + (−0.223 + 0.223i)13-s + (0.688 − 0.162i)14-s + (−0.688 − 0.724i)15-s − 0.250·16-s + (0.959 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.905 + 0.424i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0266472 - 0.119498i\)
\(L(\frac12)\) \(\approx\) \(0.0266472 - 0.119498i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (0.282 - 2.98i)T \)
5 \( 1 + (3.28 - 3.77i)T \)
7 \( 1 + (3.67 - 5.95i)T \)
good11 \( 1 + 19.5iT - 121T^{2} \)
13 \( 1 + (2.90 - 2.90i)T - 169iT^{2} \)
17 \( 1 + (-16.3 + 16.3i)T - 289iT^{2} \)
19 \( 1 + 8.66T + 361T^{2} \)
23 \( 1 + (6.73 - 6.73i)T - 529iT^{2} \)
29 \( 1 + 31.3T + 841T^{2} \)
31 \( 1 - 39.4iT - 961T^{2} \)
37 \( 1 + (25.1 - 25.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 58.9T + 1.68e3T^{2} \)
43 \( 1 + (-10.5 - 10.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (29.2 - 29.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (10.3 - 10.3i)T - 2.80e3iT^{2} \)
59 \( 1 - 42.5iT - 3.48e3T^{2} \)
61 \( 1 + 45.1iT - 3.72e3T^{2} \)
67 \( 1 + (-89.3 + 89.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 47.3iT - 5.04e3T^{2} \)
73 \( 1 + (89.3 - 89.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 41.4iT - 6.24e3T^{2} \)
83 \( 1 + (-44.9 - 44.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 4.80iT - 7.92e3T^{2} \)
97 \( 1 + (2.01 + 2.01i)T + 9.40e3iT^{2} \)
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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

−12.17586323588407403296547358487, −11.52386239321719603926406537687, −10.79477113144304650548150686823, −9.836342507817435858850761907960, −8.902862348914959203497503476192, −8.064995526637579948550625071452, −6.51114896329002528736795958747, −5.30373539737611660701303783675, −3.53480897687258985540497963497, −2.97190781841236906765952077015, 0.079973312188955936510472392629, 1.70797408133247536581158411397, 4.01048147679021818151553206336, 5.42797431257327937492730832433, 6.79835673046112983463307759421, 7.52386810972042783268270752842, 8.234969343914911072378228082819, 9.526970707733231491675168794635, 10.45706515565085305083065358051, 11.81044561696431870588304807466

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