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Properties

Label 2-210-15.14-c2-0-8
Degree $2$
Conductor $210$
Sign $0.163 - 0.986i$
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  + 1.41·2-s + (2.01 + 2.22i)3-s + 2.00·4-s + (−3.91 + 3.10i)5-s + (2.84 + 3.14i)6-s + 2.64i·7-s + 2.82·8-s + (−0.882 + 8.95i)9-s + (−5.54 + 4.39i)10-s + 6.10i·11-s + (4.02 + 4.44i)12-s − 6.09i·13-s + 3.74i·14-s + (−14.7 − 2.45i)15-s + 4.00·16-s + 7.30·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.671 + 0.740i)3-s + 0.500·4-s + (−0.783 + 0.621i)5-s + (0.474 + 0.523i)6-s + 0.377i·7-s + 0.353·8-s + (−0.0980 + 0.995i)9-s + (−0.554 + 0.439i)10-s + 0.554i·11-s + (0.335 + 0.370i)12-s − 0.468i·13-s + 0.267i·14-s + (−0.986 − 0.163i)15-s + 0.250·16-s + 0.429·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.163 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.91184 + 1.62101i\)
\(L(\frac12)\) \(\approx\) \(1.91184 + 1.62101i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (-2.01 - 2.22i)T \)
5 \( 1 + (3.91 - 3.10i)T \)
7 \( 1 - 2.64iT \)
good11 \( 1 - 6.10iT - 121T^{2} \)
13 \( 1 + 6.09iT - 169T^{2} \)
17 \( 1 - 7.30T + 289T^{2} \)
19 \( 1 - 31.7T + 361T^{2} \)
23 \( 1 + 33.8T + 529T^{2} \)
29 \( 1 + 41.0iT - 841T^{2} \)
31 \( 1 - 42.9T + 961T^{2} \)
37 \( 1 + 18.1iT - 1.36e3T^{2} \)
41 \( 1 - 36.4iT - 1.68e3T^{2} \)
43 \( 1 + 17.7iT - 1.84e3T^{2} \)
47 \( 1 - 45.3T + 2.20e3T^{2} \)
53 \( 1 + 18.1T + 2.80e3T^{2} \)
59 \( 1 - 4.22iT - 3.48e3T^{2} \)
61 \( 1 - 16.8T + 3.72e3T^{2} \)
67 \( 1 - 17.1iT - 4.48e3T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 + 105. iT - 5.32e3T^{2} \)
79 \( 1 + 74.8T + 6.24e3T^{2} \)
83 \( 1 + 76.5T + 6.88e3T^{2} \)
89 \( 1 - 26.8iT - 7.92e3T^{2} \)
97 \( 1 - 120. iT - 9.40e3T^{2} \)
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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

−12.18021693934996380304669631119, −11.58757734250018044644598698886, −10.36644902275552467319199001158, −9.656191318767828703021273944006, −8.105399972684747903180792347013, −7.50237364362889230430870042034, −5.94223633459213486988493639430, −4.65162601923298138724323866934, −3.59907768320165603549923817600, −2.56506401678705417799391168425, 1.19053581123273122018751110119, 3.10674738868592958035933662986, 4.12459473987616226743239742254, 5.57638771560213582357859093505, 6.94244084028317324273088588551, 7.78778342973116008810789757443, 8.663101504757738689108430783357, 9.917007414123309841659615258764, 11.48183569072230281600327848768, 12.04517142766975703666090529782

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