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Properties

Label 2-210-21.11-c2-0-5
Degree $2$
Conductor $210$
Sign $0.859 - 0.511i$
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 + (2.99 + 0.235i)3-s + (0.999 − 1.73i)4-s + (1.93 − 1.11i)5-s + (−3.82 + 1.82i)6-s + (−6.91 + 1.07i)7-s + 2.82i·8-s + (8.88 + 1.40i)9-s + (−1.58 + 2.73i)10-s + (17.8 + 10.2i)11-s + (3.39 − 4.94i)12-s + 8.06·13-s + (7.70 − 6.21i)14-s + (6.05 − 2.88i)15-s + (−2.00 − 3.46i)16-s + (−8.43 − 4.87i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.996 + 0.0784i)3-s + (0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.638 + 0.304i)6-s + (−0.988 + 0.154i)7-s + 0.353i·8-s + (0.987 + 0.156i)9-s + (−0.158 + 0.273i)10-s + (1.61 + 0.934i)11-s + (0.283 − 0.412i)12-s + 0.620·13-s + (0.550 − 0.443i)14-s + (0.403 − 0.192i)15-s + (−0.125 − 0.216i)16-s + (−0.496 − 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.859 - 0.511i$
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: \((2,\ 210,\ (\ :1),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.65098 + 0.453731i\)
\(L(\frac12)\) \(\approx\) \(1.65098 + 0.453731i\)
\(L(2)\) 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 \( 1 + (-2.99 - 0.235i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (6.91 - 1.07i)T \)
good11 \( 1 + (-17.8 - 10.2i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 8.06T + 169T^{2} \)
17 \( 1 + (8.43 + 4.87i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.22 + 3.84i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-36.8 + 21.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 45.5iT - 841T^{2} \)
31 \( 1 + (-1.31 + 2.26i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (16.4 + 28.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 27.8iT - 1.68e3T^{2} \)
43 \( 1 + 35.3T + 1.84e3T^{2} \)
47 \( 1 + (59.3 - 34.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (59.7 + 34.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-19.3 - 11.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (48.1 + 83.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.3 + 26.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 24.3iT - 5.04e3T^{2} \)
73 \( 1 + (63.2 - 109. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (27.9 + 48.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 106. iT - 6.88e3T^{2} \)
89 \( 1 + (23.9 - 13.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 22.2T + 9.40e3T^{2} \)
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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

−12.48304896219279636311158778996, −10.99209773796703940758398690337, −9.786011426891853094580720457426, −9.186689333658569606545575286245, −8.611626034038342185748777736369, −6.98964981022561105133724012187, −6.54849549414627939048197027037, −4.67303261718464241988891858169, −3.18947270879054241209440530385, −1.55513894213402032188712767471, 1.35912443881689004290045494078, 3.05722523314731545764837345506, 3.88149778170701285822773369605, 6.25599783508559393962788649502, 6.98400605151734604408687343540, 8.418301964127760900452500094995, 9.152455855260602783149481216146, 9.807031722216903459432153646437, 10.94498968767395182388775252071, 11.98166983058777273706068839100

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