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Properties

Label 2-210-35.18-c2-0-10
Degree $2$
Conductor $210$
Sign $0.976 - 0.215i$
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 + (−1.67 + 0.448i)3-s + (1.73 + i)4-s + (2.17 − 4.50i)5-s − 2.44·6-s + (1.63 + 6.80i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (4.61 − 5.35i)10-s + (9.66 − 16.7i)11-s + (−3.34 − 0.896i)12-s + (10.3 + 10.3i)13-s + (−0.253 + 9.89i)14-s + (−1.61 + 8.50i)15-s + (1.99 + 3.46i)16-s + (6.19 + 23.1i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.557 + 0.149i)3-s + (0.433 + 0.250i)4-s + (0.434 − 0.900i)5-s − 0.408·6-s + (0.234 + 0.972i)7-s + (0.249 + 0.250i)8-s + (0.288 − 0.166i)9-s + (0.461 − 0.535i)10-s + (0.879 − 1.52i)11-s + (−0.278 − 0.0747i)12-s + (0.799 + 0.799i)13-s + (−0.0180 + 0.706i)14-s + (−0.107 + 0.567i)15-s + (0.124 + 0.216i)16-s + (0.364 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.976 - 0.215i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.21908 + 0.241987i\)
\(L(\frac12)\) \(\approx\) \(2.21908 + 0.241987i\)
\(L(2)\) 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 \( 1 + (1.67 - 0.448i)T \)
5 \( 1 + (-2.17 + 4.50i)T \)
7 \( 1 + (-1.63 - 6.80i)T \)
good11 \( 1 + (-9.66 + 16.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.3 - 10.3i)T + 169iT^{2} \)
17 \( 1 + (-6.19 - 23.1i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-2.70 + 1.55i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-9.18 + 34.2i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 36.5iT - 841T^{2} \)
31 \( 1 + (1.49 - 2.59i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-0.422 - 0.113i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 64.3T + 1.68e3T^{2} \)
43 \( 1 + (38.9 + 38.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-10.7 - 2.88i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-29.6 + 7.95i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (34.4 + 19.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (21.4 + 37.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1.06 + 3.98i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 81.1T + 5.04e3T^{2} \)
73 \( 1 + (101. - 27.1i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (25.8 - 14.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-6.76 - 6.76i)T + 6.88e3iT^{2} \)
89 \( 1 + (-101. + 58.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (63.8 - 63.8i)T - 9.40e3iT^{2} \)
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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

−12.18653621149923408534148024617, −11.49436422657671126699630087755, −10.47608827708834975924964594950, −8.743805376129813587754518098552, −8.640097533089248202577942447028, −6.46907266443963423831496747492, −5.90473105523534661910840984625, −4.89031075400064069864832466870, −3.59064327675062804067085160368, −1.52150681072174324212917559406, 1.51519254987444798371562829155, 3.31843016575288871376800450016, 4.59947854525627351092200813267, 5.82587170910145519450461797943, 6.97330762077241383771217527957, 7.50088232697279452377841890898, 9.676005544276715648136170510351, 10.25384082768681216131246885843, 11.35954758941031122060454801176, 11.90252417635360158953008750618

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