Properties

Label 2-210-35.32-c2-0-1
Degree $2$
Conductor $210$
Sign $-0.999 - 0.0254i$
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  + (−0.366 + 1.36i)2-s + (−0.448 − 1.67i)3-s + (−1.73 − i)4-s + (1.65 + 4.71i)5-s + 2.44·6-s + (−6.62 − 2.26i)7-s + (2 − 1.99i)8-s + (−2.59 + 1.50i)9-s + (−7.05 + 0.538i)10-s + (−1.33 + 2.30i)11-s + (−0.896 + 3.34i)12-s + (−9.16 + 9.16i)13-s + (5.52 − 8.21i)14-s + (7.14 − 4.88i)15-s + (1.99 + 3.46i)16-s + (−2.70 + 0.724i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.331 + 0.943i)5-s + 0.408·6-s + (−0.946 − 0.324i)7-s + (0.250 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.705 + 0.0538i)10-s + (−0.121 + 0.209i)11-s + (−0.0747 + 0.278i)12-s + (−0.704 + 0.704i)13-s + (0.394 − 0.586i)14-s + (0.476 − 0.325i)15-s + (0.124 + 0.216i)16-s + (−0.159 + 0.0426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0254i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.999 - 0.0254i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00527955 + 0.414744i\)
\(L(\frac12)\) \(\approx\) \(0.00527955 + 0.414744i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + (0.448 + 1.67i)T \)
5 \( 1 + (-1.65 - 4.71i)T \)
7 \( 1 + (6.62 + 2.26i)T \)
good11 \( 1 + (1.33 - 2.30i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (9.16 - 9.16i)T - 169iT^{2} \)
17 \( 1 + (2.70 - 0.724i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (16.5 - 9.57i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (34.9 + 9.36i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 12.4iT - 841T^{2} \)
31 \( 1 + (-5.33 + 9.24i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (5.88 - 21.9i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 1.17T + 1.68e3T^{2} \)
43 \( 1 + (-2.70 + 2.70i)T - 1.84e3iT^{2} \)
47 \( 1 + (-19.2 + 71.7i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (2.09 + 7.83i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-93.6 - 54.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-35.0 - 60.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16.8 - 4.51i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 66.2T + 5.04e3T^{2} \)
73 \( 1 + (-9.37 - 34.9i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (83.6 - 48.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (83.4 - 83.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (-125. + 72.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (43.6 + 43.6i)T + 9.40e3iT^{2} \)
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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

−12.76786267605509706081982387534, −11.74052058844367994190161873031, −10.35446422328906737043768198005, −9.841985975028072141097995343742, −8.486101433141024080297400747403, −7.22784539732525676984169322946, −6.66291324058915958547762069603, −5.76865168793090300849699588947, −4.01034215075996263453753515869, −2.28550421017714030012988194271, 0.23404371264569285086575073358, 2.42484169538977523137473663643, 3.90980521734623947332236629009, 5.12790275210769730816035520590, 6.18491154579081437121573422503, 7.980910486614028258398879738427, 9.020026768180159602091609470592, 9.745371026243705659767108366205, 10.47898787325027655702240158225, 11.75077743181462850645468831119

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