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Properties

Label 2-210-7.6-c2-0-4
Degree $2$
Conductor $210$
Sign $0.957 + 0.288i$
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 − 1.73i·3-s + 2.00·4-s + 2.23i·5-s − 2.44i·6-s + (6.70 + 2.02i)7-s + 2.82·8-s − 2.99·9-s + 3.16i·10-s + 11.9·11-s − 3.46i·12-s − 9.67i·13-s + (9.47 + 2.86i)14-s + 3.87·15-s + 4.00·16-s + 7.09i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.447i·5-s − 0.408i·6-s + (0.957 + 0.288i)7-s + 0.353·8-s − 0.333·9-s + 0.316i·10-s + 1.08·11-s − 0.288i·12-s − 0.744i·13-s + (0.676 + 0.204i)14-s + 0.258·15-s + 0.250·16-s + 0.417i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.957 + 0.288i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.54289 - 0.375390i\)
\(L(\frac12)\) \(\approx\) \(2.54289 - 0.375390i\)
\(L(2)\) 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 \( 1 + 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 + (-6.70 - 2.02i)T \)
good11 \( 1 - 11.9T + 121T^{2} \)
13 \( 1 + 9.67iT - 169T^{2} \)
17 \( 1 - 7.09iT - 289T^{2} \)
19 \( 1 + 17.5iT - 361T^{2} \)
23 \( 1 + 2.84T + 529T^{2} \)
29 \( 1 + 13.6T + 841T^{2} \)
31 \( 1 - 19.9iT - 961T^{2} \)
37 \( 1 + 11.0T + 1.36e3T^{2} \)
41 \( 1 - 28.2iT - 1.68e3T^{2} \)
43 \( 1 + 72.9T + 1.84e3T^{2} \)
47 \( 1 - 28.3iT - 2.20e3T^{2} \)
53 \( 1 + 11.7T + 2.80e3T^{2} \)
59 \( 1 - 101. iT - 3.48e3T^{2} \)
61 \( 1 + 76.7iT - 3.72e3T^{2} \)
67 \( 1 + 76.2T + 4.48e3T^{2} \)
71 \( 1 + 95.4T + 5.04e3T^{2} \)
73 \( 1 + 120. iT - 5.32e3T^{2} \)
79 \( 1 + 14.8T + 6.24e3T^{2} \)
83 \( 1 + 60.9iT - 6.88e3T^{2} \)
89 \( 1 - 88.6iT - 7.92e3T^{2} \)
97 \( 1 - 18.8iT - 9.40e3T^{2} \)
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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

−12.04484648044353977350523276638, −11.42120930537589093460616773507, −10.48766423731344423889633203781, −8.962441542482865335425471576174, −7.88121396765609074831711477221, −6.86235054924247482897042712581, −5.86255963119773890553278950349, −4.63941748870437667970393560319, −3.14217801249409807707650257798, −1.63193564416792539393722530539, 1.71275619343256748218941584582, 3.75416009907072505150035035463, 4.56510348674573750665686664344, 5.65007309738309658333345886231, 6.95432490876856375745904315227, 8.219950207091971288194040913303, 9.259910609593212598765558861809, 10.36004721500579183452992935686, 11.58236904620423155858669601036, 11.87985967052177377108028064763

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