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Properties

Label 2-210-15.14-c2-0-9
Degree $2$
Conductor $210$
Sign $0.923 - 0.384i$
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 + (−2.70 − 1.28i)3-s + 2.00·4-s + (3.71 + 3.34i)5-s + (−3.83 − 1.82i)6-s + 2.64i·7-s + 2.82·8-s + (5.68 + 6.97i)9-s + (5.25 + 4.73i)10-s + 10.4i·11-s + (−5.41 − 2.57i)12-s − 11.9i·13-s + 3.74i·14-s + (−5.76 − 13.8i)15-s + 4.00·16-s + 29.1·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.903 − 0.429i)3-s + 0.500·4-s + (0.743 + 0.669i)5-s + (−0.638 − 0.303i)6-s + 0.377i·7-s + 0.353·8-s + (0.631 + 0.775i)9-s + (0.525 + 0.473i)10-s + 0.951i·11-s + (−0.451 − 0.214i)12-s − 0.922i·13-s + 0.267i·14-s + (−0.384 − 0.923i)15-s + 0.250·16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.923 - 0.384i$
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: \((2,\ 210,\ (\ :1),\ 0.923 - 0.384i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.00794 + 0.401107i\)
\(L(\frac12)\) \(\approx\) \(2.00794 + 0.401107i\)
\(L(2)\) 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 \( 1 + (2.70 + 1.28i)T \)
5 \( 1 + (-3.71 - 3.34i)T \)
7 \( 1 - 2.64iT \)
good11 \( 1 - 10.4iT - 121T^{2} \)
13 \( 1 + 11.9iT - 169T^{2} \)
17 \( 1 - 29.1T + 289T^{2} \)
19 \( 1 - 20.3T + 361T^{2} \)
23 \( 1 + 7.71T + 529T^{2} \)
29 \( 1 - 47.4iT - 841T^{2} \)
31 \( 1 + 35.5T + 961T^{2} \)
37 \( 1 + 58.0iT - 1.36e3T^{2} \)
41 \( 1 + 52.5iT - 1.68e3T^{2} \)
43 \( 1 - 15.7iT - 1.84e3T^{2} \)
47 \( 1 + 85.2T + 2.20e3T^{2} \)
53 \( 1 + 42.7T + 2.80e3T^{2} \)
59 \( 1 + 37.9iT - 3.48e3T^{2} \)
61 \( 1 - 53.5T + 3.72e3T^{2} \)
67 \( 1 - 27.6iT - 4.48e3T^{2} \)
71 \( 1 + 58.0iT - 5.04e3T^{2} \)
73 \( 1 - 67.8iT - 5.32e3T^{2} \)
79 \( 1 - 19.2T + 6.24e3T^{2} \)
83 \( 1 + 33.6T + 6.88e3T^{2} \)
89 \( 1 + 46.8iT - 7.92e3T^{2} \)
97 \( 1 - 65.1iT - 9.40e3T^{2} \)
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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

−12.50187519229633028029392980665, −11.35340186217906469451826849117, −10.42899329974656467259407464216, −9.662858927668091061927500086670, −7.66543815713444913398438930616, −6.95845469279271946358856126216, −5.64170613132741133077712999881, −5.27588409465921985734301878846, −3.28664089651936463820617266545, −1.71782247084584355952383265516, 1.22595378934804772589826004231, 3.49672971646216745538348362949, 4.78219611440005113824128634886, 5.67004768739232519947067877226, 6.46461585454728961690137751918, 7.939525622661138367904297364127, 9.519324128800628461052095365194, 10.11444953755672876516138350572, 11.43583910193675579751097294154, 11.93477341825700836434902648065

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