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Properties

Label 2-210-105.104-c3-0-4
Degree $2$
Conductor $210$
Sign $-0.997 + 0.0663i$
Analytic cond. $12.3904$
Root an. cond. $3.52000$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (−2.33 + 4.64i)3-s + 4·4-s + (−2.31 + 10.9i)5-s + (−4.66 + 9.28i)6-s + (−13.6 + 12.5i)7-s + 8·8-s + (−16.1 − 21.6i)9-s + (−4.63 + 21.8i)10-s − 18.9i·11-s + (−9.33 + 18.5i)12-s − 22.8·13-s + (−27.3 + 25.0i)14-s + (−45.3 − 36.2i)15-s + 16·16-s − 13.2i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.449 + 0.893i)3-s + 0.5·4-s + (−0.207 + 0.978i)5-s + (−0.317 + 0.631i)6-s + (−0.737 + 0.674i)7-s + 0.353·8-s + (−0.596 − 0.802i)9-s + (−0.146 + 0.691i)10-s − 0.520i·11-s + (−0.224 + 0.446i)12-s − 0.488·13-s + (−0.521 + 0.477i)14-s + (−0.781 − 0.624i)15-s + 0.250·16-s − 0.188i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0663i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.997 + 0.0663i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ -0.997 + 0.0663i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0362897 - 1.09307i\)
\(L(\frac12)\) \(\approx\) \(0.0362897 - 1.09307i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (2.33 - 4.64i)T \)
5 \( 1 + (2.31 - 10.9i)T \)
7 \( 1 + (13.6 - 12.5i)T \)
good11 \( 1 + 18.9iT - 1.33e3T^{2} \)
13 \( 1 + 22.8T + 2.19e3T^{2} \)
17 \( 1 + 13.2iT - 4.91e3T^{2} \)
19 \( 1 + 12.5iT - 6.85e3T^{2} \)
23 \( 1 + 76.6T + 1.21e4T^{2} \)
29 \( 1 - 19.4iT - 2.43e4T^{2} \)
31 \( 1 - 236. iT - 2.97e4T^{2} \)
37 \( 1 - 270. iT - 5.06e4T^{2} \)
41 \( 1 + 449.T + 6.89e4T^{2} \)
43 \( 1 - 230. iT - 7.95e4T^{2} \)
47 \( 1 + 30.5iT - 1.03e5T^{2} \)
53 \( 1 - 268.T + 1.48e5T^{2} \)
59 \( 1 - 123.T + 2.05e5T^{2} \)
61 \( 1 - 261. iT - 2.26e5T^{2} \)
67 \( 1 - 710. iT - 3.00e5T^{2} \)
71 \( 1 - 603. iT - 3.57e5T^{2} \)
73 \( 1 + 209.T + 3.89e5T^{2} \)
79 \( 1 - 1.10e3T + 4.93e5T^{2} \)
83 \( 1 + 694. iT - 5.71e5T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + 1.63e3T + 9.12e5T^{2} \)
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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

−12.09879345043234370243786746505, −11.61586114826846263097718912863, −10.50105142265139615823114794434, −9.838521900785875177036841332044, −8.527412702590082757934433049164, −6.90259097160385507467794035311, −6.10639535381442047452158488850, −5.05067411478845566784590818174, −3.62854289672230750650446310938, −2.78797754154919596031229707397, 0.36511501222469152523317232791, 2.00697416893194875171676599730, 3.87045609744771775931895558637, 5.06923237847627523941371969622, 6.15069150830989724560369274612, 7.21451563662090862470182692709, 8.050621719507368666779440873010, 9.533344372929891229602428681316, 10.66585957522802865390806304243, 11.92386650076653943705636695339

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