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Properties

Label 2-210-105.2-c1-0-9
Degree $2$
Conductor $210$
Sign $0.233 + 0.972i$
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 + (−1.53 + 0.806i)3-s + (−0.866 + 0.499i)4-s + (2.13 − 0.675i)5-s + (1.17 + 1.27i)6-s + (−2.45 − 0.998i)7-s + (0.707 + 0.707i)8-s + (1.69 − 2.47i)9-s + (−1.20 − 1.88i)10-s + (5.16 − 2.98i)11-s + (0.924 − 1.46i)12-s + (2.36 − 2.36i)13-s + (−0.330 + 2.62i)14-s + (−2.72 + 2.75i)15-s + (0.500 − 0.866i)16-s + (−0.914 − 0.245i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.884 + 0.465i)3-s + (−0.433 + 0.249i)4-s + (0.953 − 0.302i)5-s + (0.480 + 0.519i)6-s + (−0.926 − 0.377i)7-s + (0.249 + 0.249i)8-s + (0.566 − 0.824i)9-s + (−0.380 − 0.595i)10-s + (1.55 − 0.899i)11-s + (0.266 − 0.422i)12-s + (0.656 − 0.656i)13-s + (−0.0883 + 0.701i)14-s + (−0.702 + 0.711i)15-s + (0.125 − 0.216i)16-s + (−0.221 − 0.0594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.233 + 0.972i$
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: \((2,\ 210,\ (\ :1/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702142 - 0.553322i\)
\(L(\frac12)\) \(\approx\) \(0.702142 - 0.553322i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 - 0.806i)T \)
5 \( 1 + (-2.13 + 0.675i)T \)
7 \( 1 + (2.45 + 0.998i)T \)
good11 \( 1 + (-5.16 + 2.98i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.36 + 2.36i)T - 13iT^{2} \)
17 \( 1 + (0.914 + 0.245i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.22 + 2.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.208 + 0.0558i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.38T + 29T^{2} \)
31 \( 1 + (-1.28 - 2.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.31 + 0.352i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.15iT - 41T^{2} \)
43 \( 1 + (6.02 - 6.02i)T - 43iT^{2} \)
47 \( 1 + (-0.969 - 3.61i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.60 + 5.98i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.02 - 3.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.70 - 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.48 + 9.28i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + (12.8 + 3.43i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.874 + 0.504i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.84 - 7.84i)T + 83iT^{2} \)
89 \( 1 + (-0.188 + 0.327i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.42 - 3.42i)T + 97iT^{2} \)
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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

−12.06255698229081861645145302302, −11.04289296121228693983983217750, −10.32380687432790254715113605069, −9.411868201001657929912501853802, −8.718033444266423592348889277977, −6.57200540855633008403148877527, −6.03276911254413690725872886712, −4.52237984752078889636480042062, −3.30706874177752836872882357439, −1.02453039286179720927756328880, 1.79447877020860603972600097402, 4.20009567344754825371213217625, 5.73121031158889777226185711053, 6.53272571183587018741953355114, 6.91597662385215544013992660150, 8.717253675472278371401186048697, 9.623099464356094932634796225705, 10.44717157564998432973448238635, 11.74531936022959069246107772210, 12.63480647301932566767187203941

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