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Properties

Label 2-210-15.14-c2-0-17
Degree $2$
Conductor $210$
Sign $-0.781 + 0.624i$
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 + (1.07 − 2.79i)3-s + 2.00·4-s + (−2.52 + 4.31i)5-s + (−1.52 + 3.95i)6-s − 2.64i·7-s − 2.82·8-s + (−6.67 − 6.03i)9-s + (3.57 − 6.10i)10-s − 5.98i·11-s + (2.15 − 5.59i)12-s − 19.0i·13-s + 3.74i·14-s + (9.36 + 11.7i)15-s + 4.00·16-s + 2.02·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.359 − 0.933i)3-s + 0.500·4-s + (−0.505 + 0.863i)5-s + (−0.253 + 0.659i)6-s − 0.377i·7-s − 0.353·8-s + (−0.742 − 0.670i)9-s + (0.357 − 0.610i)10-s − 0.544i·11-s + (0.179 − 0.466i)12-s − 1.46i·13-s + 0.267i·14-s + (0.624 + 0.781i)15-s + 0.250·16-s + 0.119·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.781 + 0.624i$
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: \((2,\ 210,\ (\ :1),\ -0.781 + 0.624i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.232022 - 0.662250i\)
\(L(\frac12)\) \(\approx\) \(0.232022 - 0.662250i\)
\(L(2)\) 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 \( 1 + (-1.07 + 2.79i)T \)
5 \( 1 + (2.52 - 4.31i)T \)
7 \( 1 + 2.64iT \)
good11 \( 1 + 5.98iT - 121T^{2} \)
13 \( 1 + 19.0iT - 169T^{2} \)
17 \( 1 - 2.02T + 289T^{2} \)
19 \( 1 + 21.1T + 361T^{2} \)
23 \( 1 + 32.2T + 529T^{2} \)
29 \( 1 + 12.2iT - 841T^{2} \)
31 \( 1 - 39.3T + 961T^{2} \)
37 \( 1 + 20.3iT - 1.36e3T^{2} \)
41 \( 1 + 34.7iT - 1.68e3T^{2} \)
43 \( 1 - 40.3iT - 1.84e3T^{2} \)
47 \( 1 + 72.4T + 2.20e3T^{2} \)
53 \( 1 - 57.5T + 2.80e3T^{2} \)
59 \( 1 + 71.7iT - 3.48e3T^{2} \)
61 \( 1 - 81.7T + 3.72e3T^{2} \)
67 \( 1 - 81.7iT - 4.48e3T^{2} \)
71 \( 1 - 7.70iT - 5.04e3T^{2} \)
73 \( 1 + 12.0iT - 5.32e3T^{2} \)
79 \( 1 - 131.T + 6.24e3T^{2} \)
83 \( 1 - 115.T + 6.88e3T^{2} \)
89 \( 1 - 101. iT - 7.92e3T^{2} \)
97 \( 1 - 8.19iT - 9.40e3T^{2} \)
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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

−11.67136755657239861604058069103, −10.75256232854677076935271139203, −9.904946653873857438904619395402, −8.234024615774249446326182785232, −8.017659318592883126367309827625, −6.81898082697888167142428601290, −5.96913043065706054926168978628, −3.61966985391595385849033559158, −2.42705641556438736356632386602, −0.45150633823580773567732734530, 2.07486547617570732506023325882, 3.94179826976144648405591557122, 4.89001267332470564156068803118, 6.43633790388399429390814328210, 7.964539357975286592661265847799, 8.681657180618783688856939401370, 9.461620362639040300028164834618, 10.31080507987547202627736902227, 11.56957031462981287604667829669, 12.13029933196993474478667553179

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