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Properties

Label 2-211-211.15-c2-0-0
Degree $2$
Conductor $211$
Sign $-0.210 + 0.977i$
Analytic cond. $5.74933$
Root an. cond. $2.39777$
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  + (−2.96 + 1.71i)2-s + (−2.59 + 1.50i)3-s + (3.86 − 6.70i)4-s + 5.80·5-s + (5.14 − 8.90i)6-s + (−3.19 + 1.84i)7-s + 12.8i·8-s + (0.00637 − 0.0110i)9-s + (−17.2 + 9.94i)10-s + 3.36·11-s + 23.2i·12-s − 16.3·13-s + (6.31 − 10.9i)14-s + (−15.0 + 8.71i)15-s + (−6.46 − 11.1i)16-s + (−18.3 + 10.6i)17-s + ⋯
L(s)  = 1  + (−1.48 + 0.856i)2-s + (−0.866 + 0.500i)3-s + (0.967 − 1.67i)4-s + 1.16·5-s + (0.857 − 1.48i)6-s + (−0.455 + 0.263i)7-s + 1.60i·8-s + (0.000708 − 0.00122i)9-s + (−1.72 + 0.994i)10-s + 0.306·11-s + 1.93i·12-s − 1.25·13-s + (0.450 − 0.780i)14-s + (−1.00 + 0.580i)15-s + (−0.404 − 0.699i)16-s + (−1.08 + 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(211\)
Sign: $-0.210 + 0.977i$
Analytic conductor: \(5.74933\)
Root analytic conductor: \(2.39777\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 211,\ (\ :1),\ -0.210 + 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0259744 - 0.0321725i\)
\(L(\frac12)\) \(\approx\) \(0.0259744 - 0.0321725i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 + (105. - 182. i)T \)
good2 \( 1 + (2.96 - 1.71i)T + (2 - 3.46i)T^{2} \)
3 \( 1 + (2.59 - 1.50i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 - 5.80T + 25T^{2} \)
7 \( 1 + (3.19 - 1.84i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 - 3.36T + 121T^{2} \)
13 \( 1 + 16.3T + 169T^{2} \)
17 \( 1 + (18.3 - 10.6i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.78 - 13.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 27.9iT - 529T^{2} \)
29 \( 1 + (15.0 + 8.69i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-32.6 + 18.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (1.44 + 2.50i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (47.6 + 27.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (25.6 - 44.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (3.93 + 6.81i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (16.3 + 28.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (20.7 + 36.0i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (60.0 + 34.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + 69.7iT - 4.48e3T^{2} \)
71 \( 1 - 31.7T + 5.04e3T^{2} \)
73 \( 1 + (22.6 - 39.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 13.3T + 6.24e3T^{2} \)
83 \( 1 + (53.6 + 92.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 34.3iT - 7.92e3T^{2} \)
97 \( 1 - 86.7iT - 9.40e3T^{2} \)
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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

−12.69932880649731486148905919608, −11.43694622952043909711097090863, −10.29258011563370386914881323708, −9.931847205634125894537017337763, −9.073164964227570108830896649253, −7.936440063360609173249589963317, −6.50277029328430964355208139491, −6.06732864851204676760413923971, −4.87894757471130462026034389178, −2.08715845313197289643164946217, 0.04024174251240924736514627374, 1.53809264722429633797717234792, 2.87694934092849294535783945702, 5.23239389378194225629548970226, 6.62834396202062515385246431597, 7.30113082988927310324827111535, 8.924957936002982348752187014023, 9.601530974436118483969532251498, 10.28271164092524261550189988056, 11.42461964963241699053249411044

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