Properties

Label 2-210-35.34-c2-0-13
Degree $2$
Conductor $210$
Sign $-0.538 + 0.842i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + 1.73·3-s − 2.00·4-s + (−1.38 − 4.80i)5-s − 2.44i·6-s + (5.24 − 4.63i)7-s + 2.82i·8-s + 2.99·9-s + (−6.79 + 1.95i)10-s + 11.7·11-s − 3.46·12-s − 24.8·13-s + (−6.54 − 7.42i)14-s + (−2.39 − 8.32i)15-s + 4.00·16-s − 7.26·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.500·4-s + (−0.276 − 0.961i)5-s − 0.408i·6-s + (0.749 − 0.661i)7-s + 0.353i·8-s + 0.333·9-s + (−0.679 + 0.195i)10-s + 1.06·11-s − 0.288·12-s − 1.90·13-s + (−0.467 − 0.530i)14-s + (−0.159 − 0.554i)15-s + 0.250·16-s − 0.427·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.538 + 0.842i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.538 + 0.842i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.802606 - 1.46479i\)
\(L(\frac12)\) \(\approx\) \(0.802606 - 1.46479i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 - 1.73T \)
5 \( 1 + (1.38 + 4.80i)T \)
7 \( 1 + (-5.24 + 4.63i)T \)
good11 \( 1 - 11.7T + 121T^{2} \)
13 \( 1 + 24.8T + 169T^{2} \)
17 \( 1 + 7.26T + 289T^{2} \)
19 \( 1 + 23.0iT - 361T^{2} \)
23 \( 1 + 26.4iT - 529T^{2} \)
29 \( 1 - 57.0T + 841T^{2} \)
31 \( 1 - 10.2iT - 961T^{2} \)
37 \( 1 + 14.2iT - 1.36e3T^{2} \)
41 \( 1 - 16.2iT - 1.68e3T^{2} \)
43 \( 1 - 82.3iT - 1.84e3T^{2} \)
47 \( 1 - 51.6T + 2.20e3T^{2} \)
53 \( 1 - 64.8iT - 2.80e3T^{2} \)
59 \( 1 - 81.7iT - 3.48e3T^{2} \)
61 \( 1 + 13.1iT - 3.72e3T^{2} \)
67 \( 1 + 22.4iT - 4.48e3T^{2} \)
71 \( 1 - 91.5T + 5.04e3T^{2} \)
73 \( 1 - 71.9T + 5.32e3T^{2} \)
79 \( 1 - 45.2T + 6.24e3T^{2} \)
83 \( 1 + 17.7T + 6.88e3T^{2} \)
89 \( 1 - 77.5iT - 7.92e3T^{2} \)
97 \( 1 - 6.15T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.00170893692151189440742086599, −10.89634777648813245286795989261, −9.743865081209143295670702401703, −8.951274711486748641900730901741, −8.039056244211401328240584641134, −6.91512351124391861914396597950, −4.73483418264748281156592438196, −4.39907543218345950465812230097, −2.55230901136752530224008404718, −0.943754142650503365375538419095, 2.25483381429236291908954556076, 3.81247849052437851937096479129, 5.13281990679495077250145213429, 6.53900790632140088244546780470, 7.44552450639798129659900098150, 8.289042970581664229547470110089, 9.413026097950196329230690166680, 10.28347372648414276802028883047, 11.75254340023224633308547245496, 12.29276512698453751894853481223

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