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Properties

Label 2-210-105.62-c2-0-17
Degree $2$
Conductor $210$
Sign $0.546 + 0.837i$
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 + (1.21 + 2.74i)3-s − 2i·4-s + (−3.32 − 3.73i)5-s + (−3.95 − 1.52i)6-s + (2.29 − 6.61i)7-s + (2 + 2i)8-s + (−6.03 + 6.67i)9-s + (7.05 + 0.417i)10-s − 10.5i·11-s + (5.48 − 2.43i)12-s + (−14.9 − 14.9i)13-s + (4.31 + 8.90i)14-s + (6.20 − 13.6i)15-s − 4·16-s + (−15.4 − 15.4i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.405 + 0.913i)3-s − 0.5i·4-s + (−0.664 − 0.747i)5-s + (−0.659 − 0.254i)6-s + (0.327 − 0.944i)7-s + (0.250 + 0.250i)8-s + (−0.670 + 0.741i)9-s + (0.705 + 0.0417i)10-s − 0.961i·11-s + (0.456 − 0.202i)12-s + (−1.15 − 1.15i)13-s + (0.308 + 0.636i)14-s + (0.413 − 0.910i)15-s − 0.250·16-s + (−0.911 − 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.546 + 0.837i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.546 + 0.837i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.747247 - 0.404587i\)
\(L(\frac12)\) \(\approx\) \(0.747247 - 0.404587i\)
\(L(2)\) 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 \( 1 + (-1.21 - 2.74i)T \)
5 \( 1 + (3.32 + 3.73i)T \)
7 \( 1 + (-2.29 + 6.61i)T \)
good11 \( 1 + 10.5iT - 121T^{2} \)
13 \( 1 + (14.9 + 14.9i)T + 169iT^{2} \)
17 \( 1 + (15.4 + 15.4i)T + 289iT^{2} \)
19 \( 1 - 17.3T + 361T^{2} \)
23 \( 1 + (-23.1 - 23.1i)T + 529iT^{2} \)
29 \( 1 - 23.7T + 841T^{2} \)
31 \( 1 + 33.1iT - 961T^{2} \)
37 \( 1 + (17.6 + 17.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 11.8T + 1.68e3T^{2} \)
43 \( 1 + (22.8 - 22.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (12.6 + 12.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-15.3 - 15.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 31.0iT - 3.48e3T^{2} \)
61 \( 1 - 48.6iT - 3.72e3T^{2} \)
67 \( 1 + (77.5 + 77.5i)T + 4.48e3iT^{2} \)
71 \( 1 + 60.7iT - 5.04e3T^{2} \)
73 \( 1 + (3.52 + 3.52i)T + 5.32e3iT^{2} \)
79 \( 1 - 99.4iT - 6.24e3T^{2} \)
83 \( 1 + (16.9 - 16.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 17.6iT - 7.92e3T^{2} \)
97 \( 1 + (-34.7 + 34.7i)T - 9.40e3iT^{2} \)
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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

−11.62977519552648850496702945387, −10.89758894075004779983164520037, −9.827745979355338948380714731714, −9.005443173074745279714135369696, −7.980328898398255596717548107228, −7.35082889598979435672896920447, −5.38102256299002992945198024002, −4.63525149801214074037325148479, −3.20154590149847281091439447153, −0.52727699063809780199919785750, 1.93909392966123302319340299021, 2.90720635495984028895205840036, 4.62357986476212954630572694775, 6.64462955930385482302279485468, 7.24275601180491689261320714866, 8.391038713140010520358304527131, 9.153174233737502423650606918523, 10.39380082105130607854731031084, 11.67098452977508874564744521637, 12.04094804161693591115880244827

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