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Properties

Label 2-51-51.47-c2-0-0
Degree $2$
Conductor $51$
Sign $-0.967 - 0.253i$
Analytic cond. $1.38964$
Root an. cond. $1.17883$
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  − 0.208·2-s + (−2.91 − 0.709i)3-s − 3.95·4-s + (−5.25 + 5.25i)5-s + (0.607 + 0.147i)6-s + (2.40 − 2.40i)7-s + 1.65·8-s + (7.99 + 4.13i)9-s + (1.09 − 1.09i)10-s + (−11.4 − 11.4i)11-s + (11.5 + 2.80i)12-s − 8.12·13-s + (−0.500 + 0.500i)14-s + (19.0 − 11.5i)15-s + 15.4·16-s + (−13.5 + 10.2i)17-s + ⋯
L(s)  = 1  − 0.104·2-s + (−0.971 − 0.236i)3-s − 0.989·4-s + (−1.05 + 1.05i)5-s + (0.101 + 0.0246i)6-s + (0.343 − 0.343i)7-s + 0.207·8-s + (0.888 + 0.459i)9-s + (0.109 − 0.109i)10-s + (−1.04 − 1.04i)11-s + (0.961 + 0.234i)12-s − 0.624·13-s + (−0.0357 + 0.0357i)14-s + (1.27 − 0.772i)15-s + 0.967·16-s + (−0.798 + 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.967 - 0.253i$
Analytic conductor: \(1.38964\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1),\ -0.967 - 0.253i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0136298 + 0.105827i\)
\(L(\frac12)\) \(\approx\) \(0.0136298 + 0.105827i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.91 + 0.709i)T \)
17 \( 1 + (13.5 - 10.2i)T \)
good2 \( 1 + 0.208T + 4T^{2} \)
5 \( 1 + (5.25 - 5.25i)T - 25iT^{2} \)
7 \( 1 + (-2.40 + 2.40i)T - 49iT^{2} \)
11 \( 1 + (11.4 + 11.4i)T + 121iT^{2} \)
13 \( 1 + 8.12T + 169T^{2} \)
19 \( 1 - 24.0iT - 361T^{2} \)
23 \( 1 + (-5.74 - 5.74i)T + 529iT^{2} \)
29 \( 1 + (11.7 - 11.7i)T - 841iT^{2} \)
31 \( 1 + (-5.33 - 5.33i)T + 961iT^{2} \)
37 \( 1 + (42.7 + 42.7i)T + 1.36e3iT^{2} \)
41 \( 1 + (8.11 + 8.11i)T + 1.68e3iT^{2} \)
43 \( 1 + 2.45iT - 1.84e3T^{2} \)
47 \( 1 + 64.4iT - 2.20e3T^{2} \)
53 \( 1 - 9.84T + 2.80e3T^{2} \)
59 \( 1 - 73.8T + 3.48e3T^{2} \)
61 \( 1 + (55.3 - 55.3i)T - 3.72e3iT^{2} \)
67 \( 1 - 27.1T + 4.48e3T^{2} \)
71 \( 1 + (6.71 - 6.71i)T - 5.04e3iT^{2} \)
73 \( 1 + (1.60 + 1.60i)T + 5.32e3iT^{2} \)
79 \( 1 + (64.8 - 64.8i)T - 6.24e3iT^{2} \)
83 \( 1 + 64.2T + 6.88e3T^{2} \)
89 \( 1 - 142. iT - 7.92e3T^{2} \)
97 \( 1 + (-48.5 - 48.5i)T + 9.40e3iT^{2} \)
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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

−15.85496617699250534168990788057, −14.65497207081927693377623897313, −13.46833303990050022553604518714, −12.26600701447199569225946604776, −10.97907752688257442273658819429, −10.35350486656357174121848780663, −8.257739863971778342749493678997, −7.23323515973076952608648148754, −5.47566783326503795082671335610, −3.88225832960449190327758644504, 0.12648475512791649827639796495, 4.63674953665300073658769021198, 4.96069058168285329198338728121, 7.40251627286032629815888040473, 8.736693167935161381840105333623, 9.916317054930951956735453071015, 11.42901692726726856886134707039, 12.44898463228992838824556821540, 13.19818323646370645880585210041, 15.16698461115451395780251799223

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