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Properties

Label 2-210-35.19-c2-0-6
Degree $2$
Conductor $210$
Sign $0.964 - 0.265i$
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  + (−1.22 − 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s + (−3.86 − 3.17i)5-s − 2.44i·6-s + (6.18 + 3.28i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (2.49 + 6.61i)10-s + (−1.46 − 2.53i)11-s + (−1.73 + 2.99i)12-s + 16.5·13-s + (−5.25 − 8.39i)14-s + (1.40 − 8.54i)15-s + (−2.00 + 3.46i)16-s + (10.2 + 17.6i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s + (−0.773 − 0.634i)5-s − 0.408i·6-s + (0.883 + 0.468i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.249 + 0.661i)10-s + (−0.133 − 0.230i)11-s + (−0.144 + 0.249i)12-s + 1.27·13-s + (−0.375 − 0.599i)14-s + (0.0939 − 0.569i)15-s + (−0.125 + 0.216i)16-s + (0.600 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24414 + 0.168315i\)
\(L(\frac12)\) \(\approx\) \(1.24414 + 0.168315i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (3.86 + 3.17i)T \)
7 \( 1 + (-6.18 - 3.28i)T \)
good11 \( 1 + (1.46 + 2.53i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 16.5T + 169T^{2} \)
17 \( 1 + (-10.2 - 17.6i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.28 - 3.05i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.3 - 7.11i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 34.1T + 841T^{2} \)
31 \( 1 + (4.16 - 2.40i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-38.4 - 22.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 39.2iT - 1.68e3T^{2} \)
43 \( 1 + 50.0iT - 1.84e3T^{2} \)
47 \( 1 + (2.08 - 3.60i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (51.1 - 29.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-31.2 + 18.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (47.8 + 27.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (60.0 - 34.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 110.T + 5.04e3T^{2} \)
73 \( 1 + (47.6 + 82.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (64.7 - 112. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 5.92T + 6.88e3T^{2} \)
89 \( 1 + (85.9 + 49.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 116.T + 9.40e3T^{2} \)
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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

−11.92638037771303124734501212428, −11.17147134493366871276443344417, −10.32829102087540890248231515900, −8.920989187455192499603369619081, −8.463578562025439997367623780631, −7.63830089348629509814945948572, −5.80809831083301270990689563853, −4.46300775242772125604002226024, −3.30285024265017816831913625605, −1.35400389421203441677147324120, 1.06036076609482540331845730343, 2.97999517810658801039794042113, 4.59004387103629490913150889064, 6.24034830427735538155602065806, 7.32594704475318412240245985850, 7.912421537867740125374342500738, 8.842723925134570024013848428906, 10.19629863928708444930255175801, 11.19242132486254781235894028987, 11.74498203992655368968787177151

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