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Properties

Label 2-210-3.2-c2-0-5
Degree $2$
Conductor $210$
Sign $0.991 + 0.132i$
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  − 1.41i·2-s + (−2.97 − 0.396i)3-s − 2.00·4-s + 2.23i·5-s + (−0.560 + 4.20i)6-s − 2.64·7-s + 2.82i·8-s + (8.68 + 2.35i)9-s + 3.16·10-s + 2.01i·11-s + (5.94 + 0.792i)12-s + 11.6·13-s + 3.74i·14-s + (0.886 − 6.64i)15-s + 4.00·16-s + 17.3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.991 − 0.132i)3-s − 0.500·4-s + 0.447i·5-s + (−0.0934 + 0.700i)6-s − 0.377·7-s + 0.353i·8-s + (0.965 + 0.261i)9-s + 0.316·10-s + 0.183i·11-s + (0.495 + 0.0660i)12-s + 0.892·13-s + 0.267i·14-s + (0.0590 − 0.443i)15-s + 0.250·16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.991 + 0.132i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.991 + 0.132i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01598 - 0.0674019i\)
\(L(\frac12)\) \(\approx\) \(1.01598 - 0.0674019i\)
\(L(2)\) 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 \( 1 + (2.97 + 0.396i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64T \)
good11 \( 1 - 2.01iT - 121T^{2} \)
13 \( 1 - 11.6T + 169T^{2} \)
17 \( 1 - 17.3iT - 289T^{2} \)
19 \( 1 - 36.1T + 361T^{2} \)
23 \( 1 + 32.2iT - 529T^{2} \)
29 \( 1 - 46.1iT - 841T^{2} \)
31 \( 1 - 34.0T + 961T^{2} \)
37 \( 1 - 31.4T + 1.36e3T^{2} \)
41 \( 1 - 32.1iT - 1.68e3T^{2} \)
43 \( 1 + 51.7T + 1.84e3T^{2} \)
47 \( 1 - 92.3iT - 2.20e3T^{2} \)
53 \( 1 + 18.3iT - 2.80e3T^{2} \)
59 \( 1 - 45.4iT - 3.48e3T^{2} \)
61 \( 1 - 28.1T + 3.72e3T^{2} \)
67 \( 1 + 33.7T + 4.48e3T^{2} \)
71 \( 1 + 25.2iT - 5.04e3T^{2} \)
73 \( 1 - 48.5T + 5.32e3T^{2} \)
79 \( 1 - 32.9T + 6.24e3T^{2} \)
83 \( 1 + 82.4iT - 6.88e3T^{2} \)
89 \( 1 - 48.8iT - 7.92e3T^{2} \)
97 \( 1 + 14.5T + 9.40e3T^{2} \)
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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

−12.03765499843376012971908152884, −11.12021420885906669703727650766, −10.43323694174052335833929297830, −9.550186528746324494526371441773, −8.140772956983952642862523246028, −6.80200159880877991621805272367, −5.86368138860325867231900270146, −4.55569609414235153826339789359, −3.16904763739131390221598821674, −1.22693387444272117522431927911, 0.821742219874183493469639748733, 3.67566325990574177237358734297, 5.07210940837180629880112508447, 5.82109967185914083174411953942, 6.92219814674045483783367840885, 7.952893531935678137633014787395, 9.370179912314289405218690808358, 9.963135283950672886957466516513, 11.47609578678318871238420395153, 11.94681994564899713877434961846

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