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Properties

Label 2-210-105.17-c2-0-21
Degree $2$
Conductor $210$
Sign $0.869 + 0.493i$
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  + (−1.36 − 0.366i)2-s + (2.97 + 0.359i)3-s + (1.73 + i)4-s + (2.39 − 4.39i)5-s + (−3.93 − 1.58i)6-s + (6.65 + 2.17i)7-s + (−1.99 − 2i)8-s + (8.74 + 2.13i)9-s + (−4.87 + 5.12i)10-s + (−5.82 − 3.36i)11-s + (4.79 + 3.60i)12-s + (−3.78 − 3.78i)13-s + (−8.28 − 5.41i)14-s + (8.69 − 12.2i)15-s + (1.99 + 3.46i)16-s + (3.69 − 0.990i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.992 + 0.119i)3-s + (0.433 + 0.250i)4-s + (0.478 − 0.878i)5-s + (−0.656 − 0.263i)6-s + (0.950 + 0.311i)7-s + (−0.249 − 0.250i)8-s + (0.971 + 0.237i)9-s + (−0.487 + 0.512i)10-s + (−0.529 − 0.305i)11-s + (0.399 + 0.300i)12-s + (−0.291 − 0.291i)13-s + (−0.592 − 0.386i)14-s + (0.579 − 0.814i)15-s + (0.124 + 0.216i)16-s + (0.217 − 0.0582i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.869 + 0.493i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.869 + 0.493i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.74962 - 0.462235i\)
\(L(\frac12)\) \(\approx\) \(1.74962 - 0.462235i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (-2.97 - 0.359i)T \)
5 \( 1 + (-2.39 + 4.39i)T \)
7 \( 1 + (-6.65 - 2.17i)T \)
good11 \( 1 + (5.82 + 3.36i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.78 + 3.78i)T + 169iT^{2} \)
17 \( 1 + (-3.69 + 0.990i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-2.17 - 3.76i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.48 - 12.9i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 19.0T + 841T^{2} \)
31 \( 1 + (-0.140 - 0.0813i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.9 + 52.1i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 64.3T + 1.68e3T^{2} \)
43 \( 1 + (49.3 - 49.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-8.05 + 30.0i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (79.5 - 21.3i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (1.10 + 0.636i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (88.4 - 51.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (98.3 - 26.3i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 119. iT - 5.04e3T^{2} \)
73 \( 1 + (68.2 - 18.2i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-89.1 + 51.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (21.3 - 21.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (83.4 - 48.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (37.6 - 37.6i)T - 9.40e3iT^{2} \)
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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

−12.11725286978444410915410356366, −10.87366598137175404359823391298, −9.847636354961445747275097212613, −9.047722477459499708367734201401, −8.203377484515181437438535950566, −7.56257736912432200766727103278, −5.70248673237735534509556779090, −4.45767946164322184158381561147, −2.70475072377860867902269358602, −1.42338503599751122317360038071, 1.74604145610206970760920120199, 2.92403862088241463901180746648, 4.69244067802608064730749422143, 6.40904043620417500478027462677, 7.44438385494990918220330631593, 8.092025572724653659460555956684, 9.265452291231142478531313194703, 10.16982822095280569227442098928, 10.88870970107181962111485579792, 12.14146065235887637726323881415

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