Properties

Label 2-300-20.19-c2-0-27
Degree $2$
Conductor $300$
Sign $0.959 - 0.281i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
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  + (1.99 + 0.177i)2-s + 1.73·3-s + (3.93 + 0.707i)4-s + (3.45 + 0.307i)6-s + 1.19·7-s + (7.71 + 2.10i)8-s + 2.99·9-s + 8.22i·11-s + (6.81 + 1.22i)12-s + 11.1i·13-s + (2.38 + 0.212i)14-s + (14.9 + 5.57i)16-s − 20.9i·17-s + (5.97 + 0.533i)18-s − 27.9i·19-s + ⋯
L(s)  = 1  + (0.996 + 0.0888i)2-s + 0.577·3-s + (0.984 + 0.176i)4-s + (0.575 + 0.0512i)6-s + 0.170·7-s + (0.964 + 0.263i)8-s + 0.333·9-s + 0.747i·11-s + (0.568 + 0.102i)12-s + 0.860i·13-s + (0.170 + 0.0151i)14-s + (0.937 + 0.348i)16-s − 1.23i·17-s + (0.332 + 0.0296i)18-s − 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.67330 + 0.528391i\)
\(L(\frac12)\) \(\approx\) \(3.67330 + 0.528391i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.177i)T \)
3 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 - 1.19T + 49T^{2} \)
11 \( 1 - 8.22iT - 121T^{2} \)
13 \( 1 - 11.1iT - 169T^{2} \)
17 \( 1 + 20.9iT - 289T^{2} \)
19 \( 1 + 27.9iT - 361T^{2} \)
23 \( 1 - 9.48T + 529T^{2} \)
29 \( 1 + 40.4T + 841T^{2} \)
31 \( 1 - 55.3iT - 961T^{2} \)
37 \( 1 + 50.1iT - 1.36e3T^{2} \)
41 \( 1 + 73.6T + 1.68e3T^{2} \)
43 \( 1 - 19.0T + 1.84e3T^{2} \)
47 \( 1 - 18.0T + 2.20e3T^{2} \)
53 \( 1 + 57.2iT - 2.80e3T^{2} \)
59 \( 1 - 60.6iT - 3.48e3T^{2} \)
61 \( 1 + 21.3T + 3.72e3T^{2} \)
67 \( 1 - 9.68T + 4.48e3T^{2} \)
71 \( 1 + 68.6iT - 5.04e3T^{2} \)
73 \( 1 - 84.7iT - 5.32e3T^{2} \)
79 \( 1 + 23.2iT - 6.24e3T^{2} \)
83 \( 1 + 93.2T + 6.88e3T^{2} \)
89 \( 1 + 62.9T + 7.92e3T^{2} \)
97 \( 1 + 91.3iT - 9.40e3T^{2} \)
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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

−11.68720540841168176343920221852, −10.91720603303885375779997323881, −9.639779882505413127660428538595, −8.720560707774949075421619615760, −7.22213146785473262554284944834, −6.94092065662553611805604729006, −5.24693942174148903580707030415, −4.44470129739072604043624512078, −3.12983501676273750247927513841, −1.92258740815355644981542689314, 1.67952909197211848373730538296, 3.18041127283502089390122980813, 4.02783062180757913498251550609, 5.47819621491180948888572691507, 6.27809349551215676432842387059, 7.70196127533400712806547847646, 8.319146095446808717534579992631, 9.837138228253990890233241945781, 10.67538762642786597500739938544, 11.58432717140352504229463874194

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