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Properties

Label 2-210-35.17-c1-0-0
Degree $2$
Conductor $210$
Sign $0.519 - 0.854i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 − 0.499i)4-s + (2.23 − 0.0994i)5-s − i·6-s + (2.52 − 0.781i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.482 + 2.18i)10-s + (−1.31 + 2.27i)11-s + (0.965 + 0.258i)12-s + (1.21 + 1.21i)13-s + (0.101 + 2.64i)14-s + (−2.13 + 0.674i)15-s + (0.500 + 0.866i)16-s + (1.95 + 7.31i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 − 0.249i)4-s + (0.999 − 0.0444i)5-s − 0.408i·6-s + (0.955 − 0.295i)7-s + (0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.152 + 0.690i)10-s + (−0.395 + 0.685i)11-s + (0.278 + 0.0747i)12-s + (0.337 + 0.337i)13-s + (0.0270 + 0.706i)14-s + (−0.550 + 0.174i)15-s + (0.125 + 0.216i)16-s + (0.475 + 1.77i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.519 - 0.854i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979727 + 0.551143i\)
\(L(\frac12)\) \(\approx\) \(0.979727 + 0.551143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-2.23 + 0.0994i)T \)
7 \( 1 + (-2.52 + 0.781i)T \)
good11 \( 1 + (1.31 - 2.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.21 - 1.21i)T + 13iT^{2} \)
17 \( 1 + (-1.95 - 7.31i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.32 + 4.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.95 - 1.32i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.99iT - 29T^{2} \)
31 \( 1 + (8.66 + 5.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.02 + 3.82i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.59iT - 41T^{2} \)
43 \( 1 + (0.545 - 0.545i)T - 43iT^{2} \)
47 \( 1 + (6.12 + 1.64i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.22 + 8.28i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.86 - 6.68i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.16 + 2.40i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.47 + 0.663i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.36T + 71T^{2} \)
73 \( 1 + (13.1 - 3.53i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.78 + 4.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.99 + 7.99i)T + 83iT^{2} \)
89 \( 1 + (0.0812 + 0.140i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.35 - 4.35i)T - 97iT^{2} \)
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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

−12.85434444283291852224403409803, −11.30031008531124999883664875591, −10.53593888125773620686517771014, −9.598363437562670933846510062925, −8.524989026848482893918725889969, −7.37962779944948299140920155652, −6.25815759495283108677232834022, −5.33538223472321635525484536092, −4.29900762805221161823569372309, −1.73337544208187731407185567743, 1.42156759447870718443304892647, 2.97526938828148939816596502655, 4.98562345955490397130116379374, 5.62998428648627567478449091595, 7.14764267298488118297802564895, 8.453240913900027045088478759885, 9.364733857447104337769082233826, 10.59238070501938192477663429303, 11.04507956498584094916462654109, 12.14537350416867862583505085442

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