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Properties

Label 2-210-21.11-c2-0-10
Degree $2$
Conductor $210$
Sign $0.995 + 0.0945i$
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  + (1.22 − 0.707i)2-s + (2.17 + 2.06i)3-s + (0.999 − 1.73i)4-s + (1.93 − 1.11i)5-s + (4.12 + 0.987i)6-s + (6.85 − 1.41i)7-s − 2.82i·8-s + (0.483 + 8.98i)9-s + (1.58 − 2.73i)10-s + (−1.73 − 1.00i)11-s + (5.75 − 1.70i)12-s − 17.1·13-s + (7.39 − 6.58i)14-s + (6.52 + 1.56i)15-s + (−2.00 − 3.46i)16-s + (16.4 + 9.47i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.725 + 0.687i)3-s + (0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.687 + 0.164i)6-s + (0.979 − 0.202i)7-s − 0.353i·8-s + (0.0537 + 0.998i)9-s + (0.158 − 0.273i)10-s + (−0.157 − 0.0911i)11-s + (0.479 − 0.142i)12-s − 1.31·13-s + (0.528 − 0.470i)14-s + (0.434 + 0.104i)15-s + (−0.125 − 0.216i)16-s + (0.965 + 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.995 + 0.0945i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.995 + 0.0945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.92675 - 0.138601i\)
\(L(\frac12)\) \(\approx\) \(2.92675 - 0.138601i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-2.17 - 2.06i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-6.85 + 1.41i)T \)
good11 \( 1 + (1.73 + 1.00i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 17.1T + 169T^{2} \)
17 \( 1 + (-16.4 - 9.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (7.18 + 12.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-22.4 + 12.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 10.3iT - 841T^{2} \)
31 \( 1 + (24.0 - 41.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (21.9 + 38.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 22.8iT - 1.68e3T^{2} \)
43 \( 1 + 73.6T + 1.84e3T^{2} \)
47 \( 1 + (-20.0 + 11.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (45.3 + 26.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (43.9 + 25.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (38.9 + 67.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13.9 - 24.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 50.6iT - 5.04e3T^{2} \)
73 \( 1 + (35.5 - 61.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-17.4 - 30.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 131. iT - 6.88e3T^{2} \)
89 \( 1 + (-45.4 + 26.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 71.0T + 9.40e3T^{2} \)
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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

−12.27138646275513690476851150555, −10.97370045061752821012059724714, −10.33606813400476264829531664798, −9.288864994391687539356552588919, −8.255254816604305482950190143646, −7.09901922289738862329370360950, −5.23772457827183945675716215759, −4.72458842744016958625790430496, −3.24268635013723333822027277051, −1.89138877241810229671492299861, 1.87803646090343487269452815039, 3.12242951310928217177647375916, 4.77829161188982984097642421290, 5.89569778142987699396458894171, 7.29228355503810911017231648143, 7.77522362116120870696766980435, 9.005575059957619476350140453451, 10.11553623370612605285130753848, 11.60697704273889433323923569741, 12.26898276289108140642883560056

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