Properties

Label 2-210-105.2-c1-0-3
Degree $2$
Conductor $210$
Sign $-0.649 - 0.760i$
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 + (−1.73 + 0.0744i)3-s + (−0.866 + 0.499i)4-s + (2.11 + 0.717i)5-s + (−0.519 − 1.65i)6-s + (−1.05 + 2.42i)7-s + (−0.707 − 0.707i)8-s + (2.98 − 0.257i)9-s + (−0.145 + 2.23i)10-s + (−2.50 + 1.44i)11-s + (1.46 − 0.929i)12-s + (−4.54 + 4.54i)13-s + (−2.61 − 0.393i)14-s + (−3.71 − 1.08i)15-s + (0.500 − 0.866i)16-s + (1.39 + 0.372i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.999 + 0.0429i)3-s + (−0.433 + 0.249i)4-s + (0.947 + 0.320i)5-s + (−0.212 − 0.674i)6-s + (−0.399 + 0.916i)7-s + (−0.249 − 0.249i)8-s + (0.996 − 0.0858i)9-s + (−0.0459 + 0.705i)10-s + (−0.754 + 0.435i)11-s + (0.421 − 0.268i)12-s + (−1.26 + 1.26i)13-s + (−0.699 − 0.105i)14-s + (−0.960 − 0.279i)15-s + (0.125 − 0.216i)16-s + (0.337 + 0.0904i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.649 - 0.760i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376794 + 0.817811i\)
\(L(\frac12)\) \(\approx\) \(0.376794 + 0.817811i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0744i)T \)
5 \( 1 + (-2.11 - 0.717i)T \)
7 \( 1 + (1.05 - 2.42i)T \)
good11 \( 1 + (2.50 - 1.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.54 - 4.54i)T - 13iT^{2} \)
17 \( 1 + (-1.39 - 0.372i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.42 + 0.820i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.76 + 1.27i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 + (-0.155 - 0.269i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.82 - 0.756i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.54iT - 41T^{2} \)
43 \( 1 + (-6.09 + 6.09i)T - 43iT^{2} \)
47 \( 1 + (-1.51 - 5.65i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.138 - 0.518i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.20 - 5.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.348 + 0.603i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.216 - 0.807i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.72iT - 71T^{2} \)
73 \( 1 + (-0.180 - 0.0483i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.11 - 4.68i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.26 + 2.26i)T + 83iT^{2} \)
89 \( 1 + (-3.60 + 6.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.75 - 4.75i)T + 97iT^{2} \)
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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

−12.54501423310931935871454104860, −12.10799026683023663013180813468, −10.64214612009621353264975188964, −9.790495803748922808678331634805, −8.926951616487333535666037445816, −7.19054991029351354429287836563, −6.52589864716440998878240181666, −5.47136526282450955216648547731, −4.70136216367485154712675656704, −2.45024026344127885712515838816, 0.838322832084175022178171905412, 2.87301355483576912225013881738, 4.71779868874059571831135539985, 5.46195705156284731944779346286, 6.64578052179677127251599120101, 7.955513633646110116852682640666, 9.636716142593035688171972308106, 10.28043360427393995590757700782, 10.81044446846529625645125972115, 12.18523689507696630467463511063

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